An analysis of stroke fluid flow in a converging channel

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5-1-1985

An analysis of stroke fluid flow in a convergingchannel using a penalty function finite elementformulationKeith A. Honkala

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AN ANALYSIS OP STOKES PLUID PLOW IN A

CONVERGING CHANNEL USING A PENALTY PUNCTION

PINITE ELEMENT PORMULATION

by

Keith A. Honkala

A Thesis Submitted in

Partial Fulfillment of the

Requirements for the Degree of

MASTER OF SCIENCE

in Mechanical Engineering

Approved by:Prof. Name Illegible

(Thesis Advisor)

Requirements for the Degree of

MASTER OF SCIENCE

in Mechanical Engineering

Name Illegible(Thesis Advisor)

Prof • ---,--:-.:.=..:..:.:..;:...:...:..:..::.~~-__:_---

Approved by:

Name IllegibleProf • .-;.,;:.;;;.;...;.~~=:.:.;;;,. _

Name IllegibleProf • ---.:....;:..::..:...:..:...=...:..:..:...=.;z;.;:.:...:..=.. _

Name Illegible(Department Head)

Prof •---..,.~--__:_----=--=:..:..:...=...:..:..:...===-----

DEPARTMENT OF MECHANICAL ENGINEERING

T Kpith A. Honkala

I Keith A. Honkala prefer to be

contacted each time a request for reproduction is made. I can be

reached ·at the following address.

Date -' 7 Mfti 8r

ABSTRACT

A penalty function finite element program is developed to study

two dimensional creeping flow in a converging channel. The finite

element solution is compared to the analytical solution of purely

radial flow. The analysis reveals that the assumption of purely radial

flow is valid for small apex angles, but is inappropriate for larger

angles .

TABLE OF CONTENTS

List of Tables v

List of Figures vi

List of Symbols vii

Introduction 1

Literature Review 5

Analytical Solution to Converging Flow 8

Problem Statement 8

Solution to Converging Flow 9

Finite Element Formulation 13

Problem Statement 13

Extremization of the Functional 14

Variational Form of the Equations 17

Finite Element Equations 19

Evaluation of the Integrals 22

Choice of the Penalty Parameter 26

Mesh for the Finite Element Formulation 29

Results and Discussion 31

Sensitivity Analysis 31

Program Verification 33

Error Analysis 34

Conclusions 37

References 38

Appendices 40

A. Equations Governing Fluid Motion 40

Continuity Equation 4 0

Navier-Stokes Equation 4 0

B. Finite Element Program STOKES 42

Flow Diagram of Computer Program STOKES 4 3

Description of COMMON Block Variables 43

STOKES Data Input 4 5

Listing of STOKES 47

LIST OF TABLES

Page v

Table 1. Nominal Problem Parameters

Table 2. Mesh Variation in X-Direction

Table 3. Mesh Variation in Y-Direction

Table 4. Variation of Penalty Parameter

Table 5. Comparison of Finite Element and

Analytical Solutions

31

32

32

33

35

Page vi

LIST OF FIGURES

Figure 1. Typical Multi-Cavity Molding Machine 2

Figure 2. Visualization of Typical Converging Flow 4

Figure 3. Purely Converging Flow Idealization 4

Figure 4. Purely Converging Flow Geometry 8

Figure 5. Domain of Solution in Rectangular Coordinates 14

Figure 6. Domain of Integrals and Coordinate

Transformation 23

Figure 7. Extrapolation of Penalty Parameter 28

Figure 8. Mesh for Converging Channel Flow 30

Figure 9. Error in Neglecting Transverse Velocity 36

Page vii

LIST OF SYMBOLS

A area of integration ( cm )

Al,A2,A3 constants of integration

Bl ,B2 ,B3,B4 ,B5 constants of integration

C path on boundary of region

e element number

F a generic function

f a function; external force ( dyne)

{F} load vector ( dyne )

G a generic function

g acceleration due to gravity (cm/sec^

)

I a functional; an integral

i index of summation

unit vector in x direction

j index of summation

j unit vector in y direction

[j] Jacobian matrix

[K] stiffness matrix

L linear operator of differential equation

1 distance on boundary ( cm )

M order of Gauss integration

n unit vector normal to boundary

n number of nodes in element e

p pressure ( dyne/cirr )

p. pressure at inner radius (dyne/cm2

)

Page viii

2AP average pressure drop ( dyne/cm )

A-Panal analytically evaluated pressure drop( dyne/cm )

APj-e finite element pressure drop ( dyne/cm )

Q volumetric flow rate ( cnr/sec )

R residual of linear operator

R^ inner radius of channel ( cm )

RQ outer radius of channel ( cm )

r polar radial coordinate

S-^,S2,S3 complex constants of integration

t time ( sec )

t x component of surface traction ( dyne/cm )

ty y component of surface traction ( dyne/cm^)

{U} velocity solution vector ( cm/sec )

u x component of velocity ( cm/sec )

u approximation to velocity u ( cm/sec )

u.j. nodal approximation to u ( cm/sec )

urr component of velocity ( cm/sec )

Uqtransverse component of velocity ( cm/sec )

u_ z component of velocity ( cm/sec )

v y component of velocity ( cm/sec )

v approximation to velocity v ( cm/sec )

v.: nodal approximation to v ( cm/sec )

w z component of velocity ( cm/sec )

x rectangular coordinate

x. x coordinate of global node ( cm )

Page ix

y rectangular coordinate

Yi y coordinate of global node ( cm )

oC apex half-angle of converging channel ( rad )

If penalty parameter

rectangular coordinate

6 transverse polar coordinate

JH viscosity ( g/(cm sec) )

rectangular coordinate

mass density (gm/cm3

)

(f normal stress ( dyne/cm )

shear stress (dyne/cm2

)

0 kernel of functional

W weight function, shape function

f

Page 1

INTRODUCTION

Computer simulations of polymeric flows is becoming an increas

ingly important tool that the analyst can use to bridge the gap be

tween theoretical and empirical work. The method of finite elements

plays an important role in this analysis since it can model complex

geometries and boundary conditions. It is the intent of this study to

analyze some common assumptions made by analysts who endeavor to

simplify the mathematical problem substantially.

A typical injection molding setup is illustrated in Figure 1. In

the general case, there are almost an unlimited number of parameters

to consider in seeking an analytical solution covering the entire

system. For instance, parameters on the machine itself are flow

capacity, operation pressure and temperature, heat transfer character

istics, dimensions of runners, gates, sprues, etc.

The fluid itself could be Non-Newtonian, thus its properties could

depend on temperature, pressure, chemical reactions, and time. In

addition, the governing equations are extremely complex (see Appendix

A), so an analytical solution to the complete problem is virtually

impossible .

In an effort to understand some general aspects of the fluid

behavior, the analyst often implements assumptions and approximations

to make the mathematical problem solvable. A typical example of this

approach is the idealization of the actual flow in Figure 2 to purely

converging flow in Figure 3.

The configuration shown in Figure 2 is typical of many applica

tions in polymer processing equipment. It involves fluid travelling

through a contraction to another area of the mold, such as a conver

gent gate. The general flow illustrated here is not amenable to

Page 2

Plunger for

Pressurization

-j

i i

i '

Figure 1. Typical Multi-Cavity Molding Machine

Page 3

analytical solution [1]*. However, a number of assumptions can be

utilized to solve the purely converging flow problem of Figure 3.

Among these assumptions are that the fluid is incompressible and

Newtonian, and that the flow is steady and isothermal. The simplicity

of the resulting equations is a valuable aid in determining trends and

general behavior of flow in converging channels. However, the purely

converging flow solution becomes less applicable as a description to

the flow in Figure 2 as the apex angle (06) is increased.

It is the aim of this study to analyze the two flow problems of

Figures 2 and 3 by means of finite elements and to draw some conclu

sions about the appropriateness of the purely converging flow assump

tion. A penalty function finite element computer program STOKES is

developed in its entirety to aid the analysis. It should be noted

that finite elements is a numerical technique for solving problems,

therefore it can not exactly solve the problem. But it can model the

geometry fairly well and provide some insight into certain aspects of

the exact solution. Any comparisons in this report between analytical

and finite element solutions are made on a relative basis, as the

actual solution is not known. Experimentation is the only sure way of

truly verifying the finite element analysis.

A key parameter in this analysis will be /\P, the average pressure

drop between entrance and exit regions. This particular parameter is

important in its own right since pressure drops need to be determined

throughout a polymer molding machine to properly set and monitor

process controls, particularly in designing a"balanced"

mold [2].

However, it is also an extremely convenient parameter to use since

this one parameter sufficiently characterizes the entire solution.

* Numbers in brackets refer to citations in the Reference section.

Page 4

^ t L L L / / / i

jL < / < {/

"7 7 7 7 7 7 7

7 7 7 / / /

Figure 2. Visualization of Typical Converging Flow

Figure 3. Purely Converging Flow Idealization

Page 5

LITERATURE REVIEW

The problem of purely converging flow was solved by Hamel in 1 9 1 6

in terms of elliptic integrals. His solution was sufficiently general

that it even accounted for inertial effects in the completeNavier-

Stokes Equation. The only assumption he used was that the flow is

purely radial. His derivation is detailed in Batchelor [3], and the

problem is commonly known as"Hamel flow.

"

Morton Denn [4] includes an excellent chapter in his book covering

the various aspects of converging flow. He breaks down Hamel 's solu

tion into three cases: (1) creeping flow, for which Reynolds number

is very small and inertial effects are neglected (also known as

"Stokes flow"); (2) the lubrication approximation, which ignores the

transverse pressure gradient for small apex angles; and ( 3) the bound

ary layer approximation, useful when the inertial effects dominate at

high Reynolds numbers. These three approximations are used to effect

uncomplicated solutions and equations. While they are merely simpli

fications of the general Hamel flow problem, they are popular among

analysts and eradicate the awkward elliptic integrals. However, these

assumptions are also widely abused and misapplied, so Denn highlights

the differences of each with almost painstaking effort.

Zienkiewicz and Cheung are accredited with being the first ones to

apply finite elements to fluid problems (1965) [5], Since then numer

ous other investigators have been at the forefront of research to

apply finite elements to fluid problems: Oden, Taylor, Hood, Baker,

Olson, Strang, Fix, and Reddy, to name a few. A history of progress

Page 6

to date is included in ref. [6]. Interestingly, Zienkiewicz states

there that finite elements now dominate the field of numerical solu

tion methods for problems involving Stokes flow. This is largely due

to the fact that solving a fluid flow problem with a continuity con

straint is equivalent to solving a solids problem with a poisson ratio

near 0.5. Thus the codes existing then were modified slightly and

immediately applied to solving Stokes flow problems.

One of the first formulations to be used in fluids was the pres

sure-velocity method, where Galerkin finite elements are applied di

rectly to the governing equations. This method has drawbacks, though,

since it requires tedious assembly techniques and results in singular

matrices [7].

Another common method is the stream function formulation where the

governing equations are combined into one equation by means of stream

functions. However, this method is not extendable to three dimension

ed problems and special boundary conditions are impractical to imple

ment. The stream function-vorticity formulation suffers a similar

fate [8].

The penalty function finite element formulation is the most recent

to be applied to solving Stokes flow. It does not experience the

problems of the other formulations, but it has some numerical diffi

culties to overcome. Reddy 's book An Introduction to the Finite Ele

ment Method [9] clearly presents the penalty function method, lists a

general purpose two dimensional finite element program, and includes

example problems along with their computer output. Reddy 's book is

the basis for many of the concepts presented in this paper.

Page 7

Much research is being carried on currently to try to address the

numerical instability of the penalty method for large scale problems.

However, it has already gained wide acceptance in the solution of

Stokes fluid flow problems. Hughes et al. [10] wrote a survey paper

summarizing progress to date in using the penalty function formula

tion. The penalty method is comparatively new, and it is expected to

remain an area of active research for quite some time.

Page 8

ANALYTICAL SOLUTION TO CONVERGING FLOW

Problem Statement

The equations for purely converging flow between two inclined

plates will be developed here. Consider the flow in a converging

channel as shown in Figure 4.

Figure 4. Purely Converging Flow Geometry

The flow input is assumed to be strictly radial and inward at the

outer radius RQ. For this flow the velocities may be assumed to be ur

= u (r,6), uQ= 0, and u_ = 0 for a two dimensional problem. Equa

tions (A. 2) and (A. 6) through (A. 8) are the applicable governing

equations. Assuming that viscous terms dominate the inertial terms,

the fluid is incompressible, body forces are negligible, and the flow

Page 9

is steady, the equations governing purely converging radial flow are:

Assumed Velocities u^(r,6) Uq= 0 ( 1 )

( 2 )

( 3 )

( 4 )

( 5 )

( 6 )

( 7 )

Volumetric Flow Rate Q =

J u_ rde ( 8 )

-d

Donain of Solution Ri<r<RQ-

oi< 6 <oC

Note that Q must be stated as a negative quantity since the flow

is radially inward.

Solution to Converging Flow

The analytical solution to the system of governing equations is

straightforward. Integrating (2) gives:

ru-. = f (6) ( 9 )

Substituting this relationship into the equilibrium equations ( 3 )

Continuity Equation

1A

--2.(1^.)= 0

r "^r

b?_H hS

Equilibrium brr2 ê2

Equations <=>P

b* r de

Boundary Conditions u_.(r,oO = 0

u_.(r,-^)= 0

P(Rif0) =

Pi

Page 10

and ( 4) produces:

2r r

^P 2/4=

0f (e)

aer2

f'(6)

df

=, etc.

de

Differentiating the first equation with respect to 6 and the

second with respect to r and equating them results in an ordinary

differential equation, along with appropriate boundary conditions.

f '"(6) + 4f '(6) = 0

f(oO = 0

f(-rf) = 0

Q =

j f (6) de

This is easily solved by substituting f^(e) = Aê j into the

above differential equation, where A^ and S^ are complex constants to

3J J

be determined, and f(e) = I f-;(e). This results in Si=0, S,=H-2i,3=1 3

S^=-2i, ( i = V _1 )' so that:'3

f (6) =

Aj_ +A2e2ie

+A3e~2ie

Utilizing Euler's formula [11] ( & = cose + i sine ), produces a

solution with constants to be determined from the boundary conditions:

f (e) =

B-l + B2cos(2e) + B3sin(2e)

Page 11

The two boundary conditions on the velocities at the wall, the

volumetric flow rate equation, and the Pythagorean Theorem ( sin2e +

2cos e = 1 ) are sufficient to solve for f(e) in the following form:

sin2o<;-

sin2e

f(6) = Q _ = r ll.

sin^cos^-

o( + 2o(sin2o(

All that remains at this point is to calculate the pressure dis

tribution by integrating the equilibrium equation (3).

0P C ii "2Q cos(26)

P(r,e) = I _ dr = il drJ}r J r2

r sin c* cos <*-

d + 2^sin2^

jHQ cos(2e)

p(r,e)= + B4(e)r^

sinc^cos^-pi + 2o<sisr0(

Differentiating this expression for pressure with respect to e and

equating to Equation (4) yields B^(e) = 0 or B4(e) = Br, a constant.

This constant is determined from the boundary condition in Equation

(7). Thus the complete solution to purely radial, creeping flow in a

converging channel is:

Q sin c^-

sin2e

u_.(r,e)= ( 10 )r

sin&cos**-

^ + 2oisrrpt

Uq= 0 ( 11 )

uQ (cos2e)(R?/r2) - 1

P(r,6)=Pi+

TT (12)

R| (sintfcostf-^ +2ttsinZQC)

Page 12

Recall that Q is negative for flow that is radially inward.

Now that an analytic expression for the pressure has been derived,

the average pressure drop /\P can be computed.

/j(R0,e)R0 de ^(RêjRj de

f\ de f\ de

Ap = _______:,. .: .. ._ .

-

Thus, the final expression for the average pressure drop between

the entrance and exit of the nozzle is:

/^Q R

/\P =

5(-^

~ 1 ) sin(2o6) ( 13 )

2 R? o^ ( sin cd cos<*-<* + 2odsin2a_) rJ

Page 13

FINITE ET.FMENT FORMULATION

Problem Statement

The statement of the differential equations describing fluid flow

in rectangular coordinates is included in the appendix. The equations

are simplified substantially with the assumptions of steady, creeping

flow (ie. Stokes flow), negligible body forces, and a Newtonian,

incompressible fluid. The basic problem is to solve the equilibrium

equations and the continuity equation subjected to certain boundary

conditions, which can be specified surface tractions and/or specified

velocities. Here are the appropriate equations with the above assump

tions incorporated:

( 14 )A{ff __) + _!( a ( _ + _ ) ) -_

= o

Equilibrium ^x ^x by 2>Y ^x ^x

by by *>x ^y ^x by

bu <3V

Continuity + =0 ( 16 )

bx ^y

t-, = (-P + 2yW )n*i +^ ( + )n'j ( 17 )

Surface ^x ^y ^x

Tractions ^v >u -*v

t= (-P + 2a)n-j + U ( + )n-i^ rby

r

bY 1*

( 18 )

The domain of the solution and surface tractions are shown in

Figure 5. The surface tractions are developed by Reddy [12].

Page 14

V

<T*

dy

/

dl\<

dx \i

t,y-

i

ryx 1

Figure 5. Domain of Solution in Rectangular Coordinates

Extremization of the Functional

Numerous methods of solution are available for solving the Stoke 's

flow problem. The approach here will be to convert the basic problem

into a variational, penalty method form to which finite elements can

be conveniently applied. The advantages and disadvantages of this

particular approach are expounded upon later.

To recast the differential equations into a variational problem,

one must"guess

"the appropriate functional which, upon extremization,

yields the basic differential equations, (14) through (18).

To accomplish this, Zienkiewicz [13] proposed a functional of the

following form, where u.,=

, u^=

, etc.

3X bY

Page 15

rf i}u )t v i 3u >v

I(ux'VVx'V =jJAU()2

+ ()2

+ ( +)2

] dA

A ^x ^y 2 by 2*

-J (t^ u +

ty v) dl + -I J/ ( _ + _)2

dA ( 19 )C 2 A ^x ^y

The idea here is to modify a standard functional by adding the

continuity equation into the functional as a constraint in the least-

square sense. Gamma (6) is referred to as the penalty parameter, and

the larger the penalty parameter, the greater the constraint of the

continuity equation is enforced. The continuity equation can only be

enforced exactly when t-*vo. However, for most problems, all that

is required is ^~0( 10 ). More is mentioned later about the selection

of a suitable penalty parameter.

In general, a functional attains an extreme value when the Euler-

Lagrange equations are satisfied [14]. To verify that the proposed

functional is indeed the proper functional for this problem, the

Euler-Lagrange equations are used to extract the original differential

equations. For a functional of the form I ( u,v,ux,vl,,v ,v ) =JJ0 dA

where u=u(x,y), v=v(x,y), and 0 = 0(u,v,ux,Uy,vx,v ,x,y), the Euler-

Lagrange equations are [15]:

- A{ _ ) - A{ ) = o

û ^x ^ dy Ûy

-

-2-( ) - M ) = 0

dv ^x 2vx <5y vy

Applying these equations to the functional of Equation (19) one

Page 16

obtains two differential equations, rearranged in this useful form:

[{2a^) + y t_ ] + __L[/U_ +^ )] =0 ( 20 )^x <)x ^y dy by 3x

>. ^ v û v auAv

A[(2A+nd-+ )f -]+ i-[A<< + )] =0 ( 21 )

^y dy 3x ^x ^y ^x

A comparison of these two equations to the original equilibrium

equations (14) and (15) reveals that they are identical only if the

following relationship holds:

û :yv

P = -#( + ) ( 22 )d-x. by

where P = 0 at its itu\nimum value.

The continuity equation (16) is almost perfectly satisfied when ft

is large, so the divergence of velocity, (ux + v ), will be very

small; and pressure can be post computed from Equation (22) once the

velocities are known. Thus, velocities are the primary unknowns, and

pressure is a secondary unknown.

What has been accomplished here is the transformation of the basic

three equations (14) through (16) into two new equations (20) and (21)

which include the penalty parameter %. This was accomplished by

extremizing the functional in Equation (19). However, once this step

is completed, one need not be concerned with the functional any fur

ther. The functional is merely a means to an end, the"end"

here

Page 17

being two new differential equations. From now on the concentration

will be on the readying of the two new differential equations (20) and

(21) for a finite element solution.

Variational Form of the Equations

At this point it is appropriate to modify the newly developed

differential equations into a variational form which will lend itself

to a finite element solution. The weighted residual method [16] will

be utilized to effect the variational form of the differential equa

tions.

Let the differential equation (20) be represented by L(u,v) =

f(x,y). If u is an approximation to u and v is an approximation to v,

one can expect the residual, R = L(u,v)-

f(x,y), to be approximately

zero. However, R can not actually be zero at every point in the

region unless u=u and v=v exactly. For most purposes, it suffices to

require that the approximate solution u and v render the residual R

approximately zero in some sense.

There are several ways of implementing this requirement. The

weighted residual method multiplies the residual by a weight function

IjJ ( x,y) and integrates this product over the region of interest. It is

then required that, for a specified weight function IjJ , the integral

is equal to zero. A note about the proper form of U will be made in

the next section, but for now, it is merely a continuous function of x

and y with continuous first derivatives.

The residual of Equation (20) is ( with f (x,y) = 0 ):

Page 18

R = L(u,v) - 0 =-fl-[(2yH +Y) _ + fr _ ] + ^.[^ (_ + !_ )]dx rjx y ^y <)y x

Imposing the condition thatjj UR dA = 0 results in:

JJ { ljJ-i-[(2^+S) _ + _]+;,, A[U( + ) ] }dA = 0

A'ax d* dY

l

3y;

?3y &x

Now it is desirable to integrate by parts to reduce the order of

the derivatives of u and v. The Green-Gauss Theorem in two dimensions

is the corresponding analogy to integration by parts in one dimension.

It states [17] that, for two functions F(x,y) and G(x,y) on the region

of integration as illustrated in Figure 5:

JJ F dA = - JJ G dA +J FG n'i dl

A ^x A^x C

| F dA =

-JJ G dA +J FG n'j dl

JJA }y A^y C

Applying the Green-Gauss Theorem to the integral of the weighted

residual one gets:

JJ{.Tl(2+^+x_!1-V- + ->>*

a ^x 3x by by 2>y ^x

/-\fl ^VZ1 Ml ^V

\a[(2iA + V ) + % ] n'i dl +JWyW(_ + ) n'j dl

= 0CT

3x }y C <)y ax

Utilizing the expression for the surface tractions in Equation

(17) and recalling the fact that P = - )f( ux+vy ), a useful varia

tional form of the differential equation is obtained:

Page 19

Jfa(2^n^+^]+^+^)}dA-/utxdl = 0 (23)a>>x

'Jx ay by by ax

cT^

A similar procedure can be applied to the second differential

equation in (21):

R = L(u,v)- 0 =-E-[/t/((_ + e_) ] +A[-$_Z+ (2W + K )_ ]

ax by ax ^y ax ay

Imposing the condition that JJ UR dA = 0 results in

CC SV dv 3U >U ya pyv /,

JJ i[(2L( + ^ )+)( ] +ÎA( + ) } dA -

J Wt, dl= 0 ( 24 )

a^ ay ^x ^x 5y ^x c y

Thus the variational form of the parti al differentia] equations

has been secured. The application of finite elements immediately

follows.

Finite Element Equations

At this point the finite element approximation enters into the

solution. In finite elements, the region of interest is geometrically

divided up into a number of elements, each element containing a number

of nodes at which the approximate solution will be obtained. An

approximate solution on each element is suggested, with the under

standing that thisapproximation must satisfy the variational forms of

the differential equations obtained in the previous section. The

approximate solutions to u and v on element e are:

Page 20

n.e

ue= I Ipj(x,y)

Uj

?e

where ne is the number of nodes per element, and uând

vâre the

approximate solutions at the nodal points, ultimately to be found.

The variational form in Equation (23) becomes the following, on

element e:

A_ ^x 3=1 l* 3=1 ^Y

for each i=l , 2 , . . . ,n_

Changing the order of sumnatians :

j=i A-, d x a x a y c y

tW^_it,-i-! , ax* , Vj-/yâi . o

j-1V Jy !>x Jx Jyce'

for each i=l , 2 , . . . fne

Similarly, for the variationalform in Equation (24):

j=i a-/ ax ^y <^y^x

+ f^*^ 4- <Y+t^& ) dx ay ] Vj -/Yity

j=i a.r

ax ^x r3y ^y cej=i Ae

This represents a system of equations, and can conveniently be

for each i=l , 2 , . . . ,n.

Page 21

expressed in matrix form. The terms in the matrices [K11], [K12],

[K22], and vectors {F1} and {F2} are:

4J.jy(2/+,_-:_ftfc* + J,//__:_fiJx Jx

Ae ay ay

dx dy

Ag dy ^xAg ^x 3y

Ae^xax a-,'

ay^y

1-*

Fx=

itxdl

F2

=JV

^dl

Element

Equation:

[K1*]

[K12]T

[K12]

[K22]

{F1}

{F2}

\

V ( 25 )

JIt should be recalled that this system of equations applies only

to the particular element under consideration. One could continue

constructing stiffness matrices and load vectors for each and every

element in the finite element mesh. The element stiffness matrices

are then added (or assembled) to a global stiffness matrix in such a

way that the stiffness associated with each elemental node contributes

to the stiffness associated with each corresponding global node. Thus

one obtains a global system of equations in the form [K]{U}={F}, where

{U} represents the approximate solution at the nodal points,contain-

Page 22

ing both u^ and v^. Imposing the known essential boundary conditions

on {u} allows the solution of the system of equations for {U} [18].

The explicit form of the weight function W ( x,y) was not stated in

the previous section. Ideally, it would be continuous and smooth on

each element, capable of satisfying the boundary conditions, and allow

the calculation of a reasonably good approximation to the exact solu

tion. In addition, it should constitute a"complete"

set of functions

so a solution can be obtained which converges to the correct solution.

The shape functions Ij^, i=l,2,...ne on each element satisfy: ^= 0 at

node j ^ i or ^ x l at node j = i; j=l,2,...ne. A set of shape

functions from the Lagrange family are used in the computer program

STOKES, which is listed in the Appendix.

To recapitulate what has been done so far: the problem defined by

the system of differential equations (14) through (18) has been trans

formed into a matrix system of equations [K]{U} = {F}, where the

matrices are an assemblage of the element equations in (25). The

solution {U} of this system of equations will give an approximate

solution to the velocities at the nodes. The pressure can be post

computed from Equation (2 2) once the nodal velocities are determined.

Evaluation of the Integrals

In theory, the integralsdefined by Equations (25) can be evaluated

analytically once the shape functions IjJ are specified. However, this

is not the best way tointegrate on a computer, especially when the

elemental region of integration is irregularly shaped. A better

method involves a transformation from x,y coordinates to the set of

Page 23

master coordinates (f ,71), where the integration is carried out on a

simple square region. To be more precise, we desire to find the

transformation from (x,y) to ( < ,77 ) coordinates that will allow the

evaluation of the following three integrals on the domain illustrated

in Figure 6 (note that the function F is a generic function):

JjF{xry)dx^ , i/ _(x,y)dAx^ , and JJ (x,y)dAx^^xy

VX V*y

-1

+1

-1

-i

+1

f

Figure 6. Domain of Integrals and Coordinate Transformation

The coordinates x and y are transformed by a typical finite ele

ment approach utilizing the elemental shape functions IjJ and the geo

metric coordinates of the nodes.

n^

= I lfi(xfy) x

( 26 )

Bey= i Yi^y) y

Page 24

The partial derivatives are transformed into ( < ,V ) coordinates

through the Jacobian matrix [19]. Note that the hat (") is used to

reference a function in the ( f , * ) coordinates.

V^

= [J]s

^7

ax

aF

^y

where [J]=

ax ay

ax

Therefore,^B1aF

^F

2*

i= [jr1;ap >

V.n

)

}y

Li? *U

( 27 )

The Jacobian matrix [J] can be evaluated using the coordinate

transformations of Equation (2 6). Thus the Jacobian matrix looks like

this:

[J] =

Be* ft\ 7T

Xi1=1 *$

Ji ^x

i=i a?Yl

i=i a7Yl

( 28 )

The only remaining transformation concerns the incremental areas

dA and dA^ . Again the Jacobian matrix is useful for this purpose

[20]:

det [J] = Ijl ^dA

Page 25

( 29 )

n

At this point all transformations can be effected, and the inte

grals with which we started take on a new appearance:

1 />! -

JJ F(x,y) dA = J f f(^) | J| dff dr,

Av,, -1-1L

^

1 )F '

y-i -i

V

* jr |J| d| dâF

7 c

As these expressions stand, they still look quite formidable.

However, Gauss-Legendre integration allows rapid evaluation of the

integrals in (^ ,^ ). The idea is to approximate the integral by

evaluating the function at several well-chosen quadrature points. In

two dimensions, Gauss-Legendre integration looks like this [21]:

i a1- MM.

:/_{F(H)d5d| îj F( î' h

M M

j=

where:

I is the integral to be evaluated on the square illustrated in

Figure 6.

W- is the" weight"

assigned to each quadrature point.

f ,Jare pre-determined points at which the function is to be

1

"'evaluated, also known as"Gauss points.

"

M is the degree of integration. The larger M is the more

accurate the approximation will be to the exact value of the

integral .

Page 26

The ease of implementation and numerical efficiency makes this

method of integration very attractive in finite element work. The

weight functions and Gauss points for several degrees of integration

are listed in the computer program STOKES in the appendix.

Choice of the Penalty Parameter

At the very heart of the penalty method formulation is the penalty

parameter y, which appears in the matrix equations (2 5). As discuss

ed in the previous section, the penalty parameter measures the extent

that the continuity equation is enforced. However, some attention

must be paid to its selection or a meaningless or trivial solution can

result.

"$"

must be sufficiently large to enforce the continuity equation.

Ideally, one would like o~*^. However, this is impractical since com

puter storage (i.e. word length) has limits. In fact, when the matrix

equations are expressed in a slightly different form, it reveals how

dangerously possible a trivial solution is.

If [K]{U} = {F} is the system of equations to be solved for the

velocities {U}, the stiffness matrix [K] can be broken up into viscous

terms and penalty parameter terms (see Equations (25)) [/<[K# ] +

Y[Kv] ] {U} = {F}. Thus, in dividing by * and taking the limit:

lam [ -L[K u] + [Kv ] {U} ] = lim -1{F}= {0}

Since lim [ -[K a] {U} ] = {0} one is faced with a trivial

y-*fl> * r

solution if the matrix [Ky

] is non-singular.

Page 27

To circumvent this, Zienkiewicz [22] suggested making [Kv] singu

lar so that it is singular for any finite # . This is accomplished by

using"

reduced integration"

on the terms in [K] which depend on the

penalty parameter. For instance, if nine Gauss points are used to

evaluate [K^ ], then four Gauss points are used to evaluate [K v ].

The stiffness integrals are intentionally miscalculated so that a

singular [Kx ] matrix results. In this way lim [_L[k]{U}] = {0} while

N*

{U} {0} [23].

It is apparent that this procedure enforces the continuity con

straint only at the reduced Gauss points. Thus one would expect that

( u"x+ v ) = 0 only at these reduced Gauss points, and that its

magnitude would be greater than zero elsewhere. This has two implica

tions: ( 1 ) that a mesh with sufficient element discretization must be

made to establish the continuity condition adequately over the entire

region; and (2) that only the reduced Gauss points should be used to

post compute the pressure by Equation (22) [24].

In practice, the value of *rj need only exceed the value of the

viscosity /A, by several orders of magnitude in order to dominate the

Equations (2 5). Hughes et al. [25] has suggested the following value

for Y:

JULxlO6

<^< JJLX1012

For most problems, y ~ 1 0 is sufficient. However, this magnitude

for the penalty parameter most assuredly requires double-precision

computer storage for almost all real variables, or else the [K^]

terms will be destroyed. This storage is very costly and time con-

Page 2 8

suming. This has prompted suggesting a possible variation utilizing

extrapolation which could relax the double-precision storage require

ment.

The method utilizes the concept of "extrapolation to zero "[2 6].

Typically, a problem is solved n times, each time using a different

penalty parameter 0^, i=l,2,...,n. The values of ^ are chosen in

increasing magnitude, but kept small enough so as not to annihilate

the [K u ] terms stored in single precision. For example, one could

choose t^ = 1q1' ^2 = lo2' ^n = lon- The idea ^ to extrapolate

to 0"OO. While this can not actually be done, one can extrapolate

->0 as illustrated in Figure 7. F is the generic parameter to be

estimated accurately; it could be velocity, pressure drop, stress,

etc. F^ is the extrapolated value of F which corresponds to o-*<>0.

F' i

Fl

F2"

3

F3"

F^> ^ L

1

0 h

h

'i

h1^y

Figure 7. Extrapolation of Penalty Parameter

Page 29

Thus, one estimates the parameter F with relatively small values

of ft using single precision storage. This kind of extrapolation is

commonplace and has been successful in estimating an almost exact

solution without the benefit of an infinite mesh discretization [27].

While this method has been verified to work for medium-sized

problems ( -0(400 equations) ), single precision storage produces

dominant truncation errors in solving an extremely large system of

equations [28]. In this case the problem becomes the subject of a

trade-off study among computer run cost, amount of available computer

storage, desired accuracy of solution, sophistication of the equation

solver, and so forth.

Mesh for the Finite Element Formulation

The geometry and boundary conditions for flow in a converging

channel are illustrated in Figure 8. The symmetry in the transverse

direction enables one to cut the problem size in half.

The essential boundary conditions are as specified. Notice that

no boundary conditions are specified at the right-most edge of the

exit region. This is a region of "free-flow", and the velocity there

is not pre-determined but is resolved from the finite element solu

tion. It is also a singular point of the solution.

The flow in the entrance region is assumed to be "fully developed

flow"[2 9], and as a consequence the velocity profile there is quad

ratic with respect to y. Hence, the velocity enters in to the region

as a specified boundary condition. If the entrance region is very

Page 30

u = a quadratic function of y'v = 0

ENTRANCE

REGION

CHANNEL REGION EXIT

REGION

Figure 8. Mesh for Converging Channel Flow

long, a uniform velocity distribution matures into fully developed

flow downstream [30]. But for the sake of reducing the mesh size, the

flow is considered to be fully developed at some ndnimal distance

upstream from the converging channel.

The mesh in the channel region is uniform and is concentrated the

most there. A fairly long exit length is required, not to recover

flow that is fully developed, but to pad the wave of error emanating

from the singularity at the right exit. More will be said about this

subsequently.

Page 31

RESULTS AND DISCUSSION

Sensitivity Analysis

The limits on computer run time and storage capabilities preclude

the possibility of using an infinite penalty parameter and infinitely

discretized mesh. And neither is it necessary to do so. The extrapo

lation method described previously allows one to estimate an accurate

solution based on a series of relatively inaccurate solutions. The

method is employed in this section to determine the sensitivity of the

solution on the mesh parameters. To reiterate: these parameters have

nothing to do with the theory and/or formulation of the problem. They

are merely consequences of the approximations induced by the finite

element mesh.

The basic parameters for the reference problem are listed in Table

1. The appropriate parameters are then varied to see how sensitive

the solution is to variations in their values. In these cases, the

average pressure drop across the converging channel, Ap, is consid

ered to be the appropriate solution parameter to check sensitivity.

Table 1. Nominal Values of Problem Parameters

Kind of Elements: 9 noded quadrilaterals

Dimensions of Converging Channel:

Length L = 16 (cm)

Exit Diameter H = 2 (cm)

Apex Half-Angle oL = 45 ( ]Volumetric Flow Rate Q = 1 (air/sec)

Viscosity Constant jtL= 1 (g/(an sec))

# of Elements in x-direction = 17

# of Elements in y-direction = 5

From the very geometry of the problem one would guess that the

Page 32

discretization in the x-direction would dwarf the importance of dis

cretization in the y-direction. Indeed, this turns out to be the

case. Table 2 demonstrates how Z\P varies for various mesh schemes in

the x-direction. The extrapolation used here is similar to that

illustrated in Figure 7.

Table 2. Mesh Variation in X-Direction

NUMBER OF ELEMENTS

IN X-DIRECTION

ELEMENT LENGTH

IN X-DIRECTION

2.6667

1.6000

1.0000

PRESSURE DROP

AP

11

17

26

1.2297

1.0410

0.9353

<*> ~0. 0.7585 (extrapolated)

Since the solution parameter is extremely sensitive to variations

in the x-direction mesh, extrapolation must always be used to procure

an accurate value of /\P.

The solution parameter A.p turns out to be fairly insensitive to

variations in the y^mesh, as illustrated in Table 3.

Table 3. Mesh Variation in Y-Direction

NUMBER OF ELEMENTS ELEMENT LENGTH PRESSURE DROP

IN Y-DIRECTION IN Y-DIRECTION AP

4 0.1963 1.0424

5 0.1571 1.0410

6 0.1309 1.0402

ffQ ~0. 1.0358 (extrapolated)

Page 33

Since the increased mesh size in the y-direction is extremely

costly and Z\P is fairly insensitive to these variations, it was

decided to use a mesh with 5 elements in the y-direction for all

subsequent computer runs, and tolerate the small error introduced in

this way.

As illustrated in Table 4, there is hardly any variation in AP

for large values of the penalty parameter ^ . Hence ^= 108was

chosen with double-precision storage as sufficient for all subsequent

computer runs.

Table 4. Variation of Penalty Parameter

PENALTY PARAMETER ( t ) PRESSURE DROP ( AP)

10^ 1.040810

1.0410109

1.0410

00 1.0410 (extrapolated)

Program Verification

STOKES was run with several model problems to test the overall

accuracy and code integrity.Two problems came from Reddy 's An Intro

duction to the Finite Element Method [31]. One problem involved fluid

squeezed between two parallel plates, and the other involved the

hydrodynamic slider bearing. In both cases, the output of STOKES

compares precisely with the author's stated results.

The program STOKES was also tested with a purely converging chan

nel problem with the excellent results. However, one trend is worth

noting: the singularity atthe exit causes a wave of error to propa-

Page 34

gate upstream with further mesh discretization. Thus, the similarity

between the finite element and analytical solutions becomes progres

sively worse as one proceeds downstream [32]. In fact, if the mesh is

too fine, the error can destroy the entire solution!

One avoids this issue when solving the converging channel problem

illustrated in Figure 8. In this case the exit region engulfs the

error wave, leaving the entrance and converging channel solutions

relatively unaffected.

Error Analysis

The entrance-converging channel-exit problem of Figure 2 was

solved for various angles {cC ) using two techniques. One involved the

analytical solution of purely converging flow, i.e. ue= 0. The

second solution was obtained from a penalty function finite element

analysis on the mesh in Figure 8 . Since the assumption that Mq= 0 is

obviously in error for large oi, the finite element solution is taken

to be the accepted solution to which the analytical solution is com

pared. At this point is should be reemphasized that the finite ele

ment solution is not the exact solution to the problem. It is used on

a comparison basis with the analytical solution, and the calculated

error is a relative comparison between the two solutions.

Table 5 and Figure 9 summarize the error involved in neglecting

the transverse velocity component. Equation ( 3 0 ) is used to compute

the error, where Apanal ^ the Pressure droP in Equation (13).

Apff i*3 tîe difference between the mean pressure at the entrance and

Page 35

mean pressure at the exit of the converging channel evaluated by the

finite element program STOKES. The mean pressures are computed by

averaging the pressures at the adjacent Gauss points.

APanai-

A^% Error = x 100% ( 30 )

APFE

Table 5. Comparison of Finite Element and Analytical Solutions

AngleK )

Element

X-Length

*1A^nal APFE

ApFeiextrapolated

*4

% Error

*5

5

2.6667

1.6000

1.0000

7.0630

7.1196

7.0977

7.0896

7.0702 -0.10

15

2.6667

1.6000

1.0000

2.6444

2.8383

2.7689

2.7395

2.6769 -1.21

25

2.6667

1.6000

1.0000

1.5034

1.8411

1.7249

1.6700

1.5638 -3.86

35

2.6667

1.6000

1.0000

0.9605

1.4358

1.2791

1.1979

1.0528 -8.76

45

2.6667

1.6000

1.0000

0.6324

1.2297

1.0410

0.9353

0.7585 -16.62

*1 Table 2

*2 Equation (13)

*3 Computer run of STOKES for problem in Figure 8

*4 Table 1

*5 Equation (30)

Faf-e 36

!-l

OU

c0)

os-l

30"

28-

26-

24.

22.

20-

15-

16-

14-

12-

'ZcL

1

r

5 10 15 20 25 30 35 40 50 55 ou

HALF -APEX ANGLE OF CONVERGING CHANNEL <# )

Figure 9. Error in Neglecting Transverse Velocity

Page 3 7

CONCLUSIONS

The pressure drop across a converging channel has been evaluated

in two ways: (1) analytically using the assumption of purely radial

flow; and (2) using the finite element program STOKES, modelling the

converging channel with entrance and exit regions. The comparison

between solutions indicates that the purely radial flow assumption is

quite valid for small apex angles, but is inappropriate for larger

angles .

The penalty function finite element formulation has been verified

to be a useful tool in solving Stokes fluid flow problems. By elimi

nating pressure terms from the stiffness matrix, the size of the

resulting system of equations is minimal. The boundary conditions are

easily implemented. Attention must be paid to the value of the penal

ty parameter and the mesh discretization employed, since changes in

these can cause the solution to vary considerably. Extrapolation

schemes are useful in aiming for the exact solution.

In using the penalty function finite element method, considerable

error is introduced at a free surface. To compensate for this, one

must keep the free surface far enough away from the region of interest

so as to leave it quite unaffected.

As with any finite element analysis, caution should be excercised

in using theseresults for design purposes. Being a numerical method,

it does not give an exact solution. Its real practicality lies in its

ability to guide experimental procedures for complex problems. Any

finite element analysis should be followed up by a good test program.

Page 38

REFERENCES

[1] Morton M. Denn, Process Fluid Mechanics (Englewcod Cliffs, N.J.:

Prentice-Hall, 1980), p. 211.

[ 2 ] StanleyMiddleman, Fundamentals of Polymer Processing ( NewYork:

McGraw-Hill, 1977), p. 264.

[3] G. K. Batchelor, An Introduction to Fluid Dynamics (New York:

Cambridge University, 1 9 6 7 ), pp. 2 1 7-2 1 9, 2 9 4-3 0 2 .

[4] Denn, chapter 10.

[5] 0. C. Zienkiewicz & Y. K. Cheung, "Finite Elements in the Solu

tion of Field Problems"

, The Engineer (1965), pp. 507-510.

[6] 0. C. Zienkiewicz, et al., editors, Finite Elements in Fluids,vol. 5 (New York: Wiley, 1984), pp. 1-5.

[7] J. N. Reddy, An Introduction to the Finite Element Method (New

York: McGraw-Hill, 1984), pp. 281-28 3.

[8] C. Taylor and P. Hood, "A Numerical Solution of the Navier-

Stokes Equations Using the Finite Element Technique"

, Computers

& Fluids (London: Pergamon, 1973), pp. 81-85.

[9] ,An Introduction to the Finite Element Method, pp. 283-292,358-374.

[10] T. R. Hughes, W. K. Liu, A. Brooks, "Review of Finite Element

Analysis of Incompressible Viscous Flows by the Penalty FunctionFormulation,"

Journal of Computational Physics (1979), vol. 30,

no. 1, pp. 1-60.

[11] J. D. Paliouras, Complex Variables for Scientists and Engineers

(New York: Macmillan, 1975), p. 82.

[12] , An Introduction to the Finite Element Method, pp. 267,

281.

[13] 0. C. Zienkiewicz & P. N. Godpole, "Viscous, Incompressible Flow

with Special Reference to Non-Newtonian ( Plastic) Fluids"

, Fi

nite Elements in Fluids, vol. 1 (New York: Wiley, 1978), p. 37.

[14] J.N.Reddv.Energy and Variational Methods in Applied Mechanics

(New York: Wiley, 1984), p. 108.

[15] I. M. Gelfand & S. V. Fomin, Calculus of Variations (Englewood

Cliffs, N.J.: Prentice-Hall, 1963), p. 181.

Page 39

t16] ,Energy and Variational Methods in Applied Mechanics,p.212.

~ " ^

[17] S. S. Rao, The Finite Element Method in Engineering (New York:

Pergamon, 1982), pp. 618, 619.

[18J , An Introduction to the Finite Element Method, pp. 66-79.

[19] H. F. Davis & A. D. Snider, Introduction to Vector Analysis, 4thed. (Boston: Allyn and Bacon, 1979), p. 266.

[20] Ibid, p. 271.

[21] R. L. Burden, J. D. Faires, A. C. Reynolds, Numerical Analysis,2nd ed. (Boston: Prindle, Weber & Schmidt, 1981), pp. 167-170.

[22] 0. C. Zienkiewicz, The Finite Element Method, 3rd. ed. (London:

McGraw-Hill, 1977), pp. 287-2 94.

[23] , An Introduction to the Finite Element Method, p. 284.

[24] Ibid., p. 285.

[25] T. R. Hughes, et al., pp. 13,18.

[26] , The Finite Element Method, pp. 34-35.

[27] Ibid.

[28] R. L. Burden, et al., pp. 398-402.

[ 29 ] Robert Fox &Alan McDonald, Introduction to Fluid Mechanics ( New

York: Wiley, 1978), p. 330.

[30] Denn, p. 211.

[31] , An Introduction to the Finite Element Method, pp. 35 8-

367.

[32] Graham F. Carey & J. Tinsley Oden, Finite Elements, vol. 3

(Englewood Cliffs, N. J.: Prentice-Hall, 1984) pp. 37-4 0.

Page 4 0

APPENDIX A. EQUATIONS GOVERNING FLUID MOTION

The governing equations describing fluid flow are presented here

in an effort to provide a complete document. All of the equations are

taken from Principles of Polymer Processing, Tadmor and Gogos (New

York: Wiley, 1979), pages 108 and 118.

Continuity Equation

The continuity equation expresses the concept that the flow field

is a continuous function with no holes or gaps in the solution. That

is, at every point in the flow field no mass is created or destroyed;

it is merely transported from one region to another.

RECTANGULAR COORDINATES (x,y,z):

_^-+_(/>u) +-i-(pv) +-i-(/)w) = 0 (A.l)

dt d* I &Y ' bz '

CYLINDRICAL COORDINATES (r ,0 , z ) :

^. + __^.(prur) +-(DUq) +-^-(pu_) = 0 (A.2)

dt r dr I rê ' d* '

Navier-Stokes Equation

This equation is the marriage of two somewhat independent con

cepts: ( 1) the conservation of momentum, usually expressed in terms of

a force balance among the stresses; and (2) that there exists a rela

tionship between fluid stresses and velocities involving certain mate

rial constants.

Page 41

RECTANGULAR COORDINATES (x,y,z):

du du bu bu b? ^2u A A0( +u + v_ +w ) = -

_ + U ( +0

+0

j + (A>3)' &t dx y Jz ^x

^ ^x2 by2 *z2 I

^v ^v ^v &p fr & \*

p(_ +u + v- +w ) = -

_ + y< +6

+ ) +p g*t ^x y ^z aY

d*2

V^z2 '

(A. 4)

>w >w :*w "?iw 7)? ?fa & "&

9{L + xL + j_ + ^)=-__+A(i^

+ 1^ + 1.) + /)gz (A.5)

^t $x }y ^z ^z^x2 ly2 ?z2 f

CYLINDRICAL COORDINATES (r,6,z):

f^t dr r ^0 br dz (A.6)

^P, . r

a,1

a, t

xa2%

2 ape a2ur

^

+ in[(-(nv)) + i^__-_^_ + __ ] +/)g^rr^r

^r2

^62 r2 2e 2

z2 '

+ U^+ + + U_. )

d t dr r ^6 r ^z (A-7)

rêA^r r )r

r2 de2 r2 be hz2 \

duz ûz "6 duzu Ûz,

p(^_ + u- + + uz-

I t 3r r ^6 5 z (A.8)

Jz^

r^r 3rr2 ^62 ^z2

"

Page 42

APPENDIX B. FINITE ELEMENT PROGRAM STOKES

STOKES is a general purpose penalty function finite element prog

ram to solve two dimensional Stokes fluid flow. It is sufficiently

general as it allows complex geometries and boundary conditions. The

program is written completely in modular form, as extensive use is

made of subroutines and structured programming. Brief comments are

provided inside the program to aid the reader's perception of what is

taking place. The flow of STOKES is similar to Reddy [ref. 9], while

the variable names and subroutine structure follow Carey & Oden [ ref.

32, vol. 1 ].

To take advantage of the similar geometries involved in solving

converging flow problems, several mesh generation schemes are in

cluded. The description of the data input and program structure are

provided in the next few pages.

Flow Diagram of Computer Program STOKES

NOTE: All subroutines return to their calling program once completed.

Flow of main program:

Page 4 3

Flow of subroutines:

PREP-MESHGEN

PROC -SHAPE JACOBI

POST

-SHAPE 'JACOBI

PERROR

Description of COMMON Block Variables

vXMMDN/CCON/

TITLE

NNODE

NELEM

NBC

NPT

Control parameters

Problem title

Total number of nodes

Total number of elements

Number of essential boundary conditions

Number of point loads

CCMMON/CNODE/

GX( )

GY( )

Global node data

X-coordinates of nodes

Y-coordinates of nodes

Page 44

CX3MM0N/CELEM/KHJD( )

NODES( , )

FX

FY

CQMMON/CMATRX/HBAND

GU( )

GK( , )

GF( )

EK( , )

EF( )

CQMMON/CMAT/MU

GAMMA

COMMON/CINT/NINT

XIQ( , )

WT( , )

CCMMON/CBC/

NCDEBC( )

DIRBC( )

VALBC( )

CCMMON/CPT/

NODEPT( )

DIRPT( )

VALPT( )

COMMON/CPRINr/

PRT( )

CCMMON/CALT/

ALT

L

H

Q

ALPHA

EX

EY

Element data

Number of nodes per element

Connectivity matrix between element and

global nodes

Body force in X direction

Body force in Y direction

Matrices

Half bandwidth of global stiffness matrix

Solution vector containing velocitiesGlobal stiffness matrix (half band form)Global load vector

Element stiffness matrix

Element load vector

Material parameters

Viscosity coefficient

Penalty parameter

Gauss integration parameters

Order of integration

Location of Gauss points

Integration weights

Essential boundary condition data

Global node of boundary condition

Direction of boundary condition

Magnitude of boundary condition

Point load data

Global node of point load

Direction of point load

Magnitude of point load

Printout control parameters

Printer control parameters

Specific problem description parameters

Signifies which error routine to use

Length of nozzle

Half-height of nozzle exit

Volumetric flow rate

Half angle of nozzle

Number of elements in X direction

Number of elements in Y direction

STOKES Data Input

Page 4 5

DATA CARDS

TITLE

PRT(1),PRT(2),...,PRT(10)

DESCRIPTION

NNODE,NELEM,NBC,NPT,MESH,KIND( 1 )

MU,GAMMA

Problem description, 80 characters

Printer controlparameters (1=Y,0=N)

l=print element stiffness matrix [EK]2=start at elem

3=end at elem

4=print global stiffness matrix [GK] ;

5=start at equation

6=end at equation

7^?rint problem data?

8=print nodal solution (velocities)?

Control parameters:

NNODE=number of global nodes

NELEM=number of elements

NBC==number of specified BC's

NPT=3iumber of specified forces

MESH=^nesh generation parameter:

0=no mesh generation

l=mesh generation for radial flow in

Nozzle with 4-node quadrilaterals

2=mesh generation for radial flow in

Nozzle with 9-node quadrilaterals

3=mesh generation for Nozzle/Pipe

with 9-node quadrilaterals

KIND=degree of quadrilateral

4=four node bilinear quadrilateral

8=eight node"serendipity"

quad.

9=îine node isoparametric quad.

Material parameters:

MU=viscosity constant

GAMMA=penalty parameter; note that

GAMMA=MU*1.E8, approximately

WITHOUT MESH OTJERATCON

GX(1),GY(1)

GX(2),GY(2)

GX(NNODE) ,GY(NNODE)

FX,FY

Gecmetric coordinates of global nodes

Body forces, assumed uniform

Page 46

N0DES(1,1) ,N0DES(1,2) .. .

N0DES(2,1) ,NODES(2,2) . . .

NODES (NELEM, 1 ) ,NODES (NELEM, 2 ) . ,

NODES (ELEM, . . . )=Connectivity matrix

containing global nodes of element

NODEBC ( 1 ) ,DLRBC ( 1 ) ,VALBC ( 1 )

NODEBC(2) ,DIRBC(2) ,VALBC(2)

NODEBOglobal node of B.C.

DIRBC=direction of action of B.C.

1=X direction

NQDEEC(NBC),DIRBC(NBC),VALBC(NBC) 2=Y direction

.VAIiBC=nagnitude of B.C.

WITH MESH GENERATION

EX ,EY ,ALT ,L ,H ,Q ,ALPHA

FX,FY

Specific problem parameters:

EX==number of elements in X-direction

Note that EX=11,14,17, . . .when MESH=3

EY=number of elements in Y-direction

ALT indicates if known solution exists

0=solution is unknown

l=solution for radial flow in nozzle

2=solution for flow in nozzle/pipe

I>length of nozzle

H=height of nozzle exit

Q=volumetric flow rate

ALPHA=half-angle of channel, in deg.

Body forces, assumed uniform

NODEPT ( 1 ) ,DIRPT ( 1 ) ,VALPT ( 1 )

NODEPT ( 2 ) ,DIRPT ( 2 ) ,VALPT ( 2 )

NODEPT(NPT) ,DIRPT(NPT) ,VALPT(NPT)

-999

NODEPT=global node of point load

DIRPT=direction of action of load

1=X direction

2=Y direction

Signifies the end to the data

NOTES:

1. Any problem can be input by knowing nodal coordinates, connectiv

ity matrix, etc. by using MESH=0 in data deck. However, to take

advantage of symmetry and known solutions for large problems,

mesh generation and error routines should be utilized. The main

program must be modified in the case of general problems for

MESHGEN and PERROR subroutines. However, as the program stands

there are three different mesh generators to cover the Nozzle and

Nozzle/Pipe problems. Specifying ALT=1,2,... calls the error

subroutine PERROR.

2. The input data is completely free-format. However, some input

variables sit on the same line, as indicated in the left-hand

column above.

PROGRAM STOKES

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA* PROGRAM STOKES K. HONKALA JAN '85

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

C FINITE ELEMENT PROGRAM TO SOLVE 2-DIM STOKES FLUID FLOW. ISOPARAMETRIC

C RECTANGULAR ELEMENTS ARE USED.

CHARACTER*80 TITLE

INTEGER NNODE, NELEM,KIND,NODES,NINT,NBC,NODEBCDIRBC,EX,EY,+ NODEPT,DIRPT,NPT,HBAND,PRT,ALTREAL*B GX,GY,GK,GF,GU,EK,EF,MU, GAMMA,XIQ,WT,FX,FY,VALBCVALPT,

+ L,H,Q,ALPHA

COMMON /CCON/TITLE ,NNODE , NELEM , NBC ,NPT

COMMON/CNODE/GX( 700 ) ,GY( 700 )

COMMON/CELEM/KIND(400),NODES(400,9),FX,FYCOMMON/CMATRX/HBAND,GU(1400) ,GK(1400,90) ,GFU400) ,EK( 18,18) ,EF(18)

COMMON/CMAT/MU, GAMMA

C0MM0N/CINT/NINT,XIQ(5,5) ,WT(5,5)

COMMON/CBC/NODEBC(200),DIRBC(200),VALBC(200)

COMMON /CPT/NODEPT( 50 ) ,DIRPT( 50 ) ,VALPT( 50 )

COMMON /CPR INT/PRT (10)

COMMON/CALT/ALT,L,H,Q,ALPHA,EX,EY

C

C CONSTRUCT FINITE ELEMENT MESHGEOMETRY ONLY

C NOTE THAT *PREP* CALLS SUBS RDATA,MESHGEN,PDATA,SETINT

CALL PREP

C COMPUTE ELEMENT MATRICES AND SOLVE EQUATIONS

C NOTE THAT *PROC* CALLS SUBS ELEMENT, SHAPE, JACOBI,ASSEMB,APLYBC,GAUSSEL

CALL PROC

G POSTPROCESS TO OUTPUT RESULTS

C NOTE THAT *POST* CALLS SUBS PNODE,PELEM, SHAPE, JACOBI,PERROR

CALL POST

C

ENDAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

SUBROUTINE PREPAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA:

C PREPROCCESSOR SUBPROGRAM WHICH CALLS INPUT/DATA SUBROUTINES

INTEGER TEST

C

C READ IN DATA FOR PROBLEM DESCRIPTION:

CALL RDATA

C

C TEST TO INSURE ALL THE DATA HAS BEEN ENTERED:

READ(5,*) TEST

IF (TEST .NE. -999) THEN

WRITE(6,*>'

ERROR IN DATAINPUT'

STOP

END IF

C

C OUTPUT DATA FOR PROBLEM DESCRIPTION:

CALL PDATA

C

C SET POINTS AND WEIGHTS FOR GAUSS INTEGRATION:

CALL SETINT

C

RETURN

END

aaÂAAAAAAAAAAAAAAAAA************************************************************

SUBROUTINE RDATAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

C INPUT DATA FOR PROBLEM DESCRIPTIONCHARACTER*80 TITLE

INTEGER NNODE,NELEM,KIND,NODES,NBC,NODEBCDIRBC,NODEPT,DIRPT,+ NPT,PRT,NODE,ELEM, BC,POINT,MESHREAL*8 GX,GY,MU,GAMMA,FX,FY,VALBCVALPT

COMMON/CCON/TITLE,NNODE,NELEM,NBC,NPTCOMMON /CNODE/GX( 700 ) ,GY( 700 )

COMMON/CELEM/KIND( 400 ) ,NODES (400,9) ,FX,FYCOMMON /CMAT /MU , GAMMA

COMMON/CBC/NODEBC<200) ,DIRBC(200) ,VALBC(200)COMMON/CPT/NODEPT(50),DIRPT(50) ,VALPT(50)COMMON /CPRINT/PRT (10)

C PRT(I)=PRINTER CONTROL PARAMETERS: I=1:ELEMENT PRINTOUT (1Y,0=N> 2:STARTC AT ELEMENT # 3:END AT ELEMENT # 4.-GL0BAL PRINTOUT (1=Y,0=N) 5:START

C AT EQUATION # 6:END AT EQUATION # 7 .-PRINT OUT DATA (1=Y,0=N)

C 8:VELOCITY SOLUTION PRINTOUT (1=Y,0=N)C

C READ IN CONTROL AND PRINT OUT PARAMETERS:

READ(5,20) TITLE

20 FORMAT(A80)READ(5,*) (PRT(I),I=1,10)

READ(5,*) NNODE,NELEM,NBC,NPT,MESH,KIND(l)READ(5,*) MU,GAMMA

C

IF (MESH .EQ. 0) THEN

C INPUT GLOBAL GEOMETRIC COORDINATES OF GLOBAL NODES:

DO N0DE=1,NNODE

READ(5,*> GX ( NODE ),GY(NODE)

END DO

C INPUT ELEMENT DATA:

READ(5,*) FX,FY

DO ELEM=1,NELEM

KIND(ELEM)=KIND(1)

READ(5,*) (NODES (ELEM,NODE) ,N0DE=1,KIND(ELEM) )

END DO

C INPUT ESSENTIAL BOUNDARY CONDITIONS:

DO BC=1,NBC

READ(5,*) NODEBC(BC),DIRBC(BC),VALBC(BC)

END DO

ELSE

IF (MESH .EQ. 1) CALL MESH4

IF (MESH .EQ. 2) CALL MESH9

IF (MESH .EQ. 3) CALL MESHNOZ

END IF

C

C INPUT POINT LOADS:

DO POINT=l,NPT

READ! 5,*) NODEPT(POINT),DIRPT(POINT),VALPT(POINT)

END DO

C

RETURN

ENDAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

SUBROUTINE PDATA

AAAAAAAAAAAAAAAA AA*************************************** ************************

C OUTPUT DATA OF PROBLEM DESCRIPTION

CHARACTER* 80 TITLE

INTEGER NNODE,NELEM,KIND,NODES,NBC,NODEBCDIRBC,NODEPT,DIRPT,

+NPT,NODE,ELEM,BC,POINT,PRT,POS

REAL*B GX,GY,MU,GAMMA,FX,FY,VALBC,VALPT

COMMON/CCON/TITLE, NNODE,NELEM,NBC,NPTCOMMON /CNODE/GX( 700 ) ,GY( 700 )

COMMON/CELEM/KIND(400) ,NODES(400,9) ,FX,FYCOMMON /CMAT/MU , GAMMA

COMMON /CBC/NODEBC ( 200 ) ,DIRBC ( 200 ) ,VALBC ( 200 )

COMMON/CPT/NODEPT( 50 ) ,DIRPT( 50 ) ,VALPT( 50 )

COMMON/CPRINT/PRT( 10 )

C

WRITE(6,*) ( -A************** PROGRAM STOKES TO SOLVE 2-DIM

+STOKES FLOW aaaa**aaaaaaaaa)

WRITE(6,10)

10 F0RMAT('0'

,'********** CONTROL PARAMETER OUTPUT **********)

WRITE(6,20)TITLE

20 FORMATCO', 'TITLE: *,A70)

WRITE(6,*)'

TOTAL NUMBER OF NODES IN MESH'

,NNODEWRITE(6,*)'

TOTAL NUMBER OF ELEMENTS IN MESH'

,NELEMWRITE(6,*)'

NUMBER OF SPECIFIED B.C.S'

,NBC

WRITE(6,*)'

NUMBER OF SPECIFIED BOUND. FORCES'

,NPT

WRITE<6,*)'

VISCOSITY CONSTANT (MU)'

,MU

WRITE(6,*)'

PENALTY PARAMETER ',GAMMA

C

IF (PRT(7) .EQ. 0) RETURN

WRITE(6,25)

25 FORMATCO','********** NODAL DATA OUTPUT aaaaaaaaaa')

WRITE(6,2B)

28 FORMAT ('

0'

, 2 (*

NODE X-COOR Y-COOR'

) )

DO N0DE=l,(NN0DE+l)/2

POS=NODE+(NNODE+l ) /2

WRITE(6,30)N0DE,GX(N0DE) ,GY(NODE) ,POS,GX(POS) ,GY(POS)

30 F0RMAT(1X,2(I4,4X,E12.5,4X,E12.5,7X))

END DO

C

WRITE(6,80)

80 FORMAT*'0',**********ELEMENT DATA OUTPUT **********')

WRITE(6,82) FX,FY

82 FORMATt'0'

, 'BODY FORCES FX,FY:'

,2(4X,E12.5) )

WRITE(6,85)

85 FORMAT('0'

, 'ELEMENT GLOBAL NODES'

)

DO ELEM=1,NELEM

WRITE(6,90) ELEM, (NODES(ELEM,NODE) ,N0DE=1,KIND(ELEM) )

90F0RMAT(1X,I4,8X,<KIND(ELEM)XI3,'

'))

END DO

C

WRITE(6 100)

100 FORMATt'0','**********BOUNDARY CONDITION DATA **********)

WRITE(6,110)

110 F0RMAT('0',2C NODE DIRECTION VALUE *))

DO BC=l,(NBC+l)/2

POS=BC+(NBC+l)/2

WRITE(6,120) NODEBC(BC) ,DIRBC(BC) ,VALBC(BC) ,NODEBC(POS) ,

+ DIRBC ( POS ) ,VALBC ( POS )

120 F0RMAT(1X,2(I4,5X,I2,5X,E12.5,12X))

END DO

C

RETURN

ENDAA*AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA*AAAAAAAAAAAAAAAAAAA

SUBROUTINE SETINT

AAAAA****************************AA*A****''t,','''A'''',t'''''','''''******,',**''tA******''''''***A*******

C SETS VALUES OF THE INTEGRATION PARAMETERS USED IN GAUSSIAN INTEGRATION

INTEGER NINT

REAL*8 XIQ,WT

C0MM0N/CINT/NINT,XIQ(5,5),WT(5,5)C ONE POINT QUADRATURE

XIQ(1,1)=0.0

WT(1,1)=2.0

C TWO POINT QUADRATURE

XIQ(2,1)=-1.0/SQRT(3.0)

XIQ(2,2)=-XIQ(2,1)

WT(2,1)=1.0

WT(2,2)=WT(2,1)

C THREE POINT QUADRATURE

XIQO,1)=-SQRT<3.0/5.0)XIQ(3,2)=0.0

XIQO,3)=-XIQO,l)

WTO, 1)=5. 0/9.0

WTO, 2)=8. 0/9.0

WT(3,3)=WT(3,1)

C

RETURN

ENDA*****AA*******AAAA****A****AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAii

SUBROUTINE MESH4AAAAAAAAAAAAAAAAAAAAAAAAA********************************************************

C GENERATE A MESH OF 4-NODED ELEMENTS. MESH IS GENERATED ON A POLAR

C REGION.

INTEGER KIND,NODES,NODEBCDIRBC,NX,NY, I,J,K,NODE,TOP,B0T,ALT

REAL*8 FX,FY,GX,GY,VALBC,L,H,Q,ALPHA,RO,THETA,XA,YA,XB,YB,VAV+ A,B,DELA,DELB,AREA,X,F

COMMON/CELEM/KIND(400) ,NODES(400,9) ,FX,FY

COMMON/CNODE/GX(700),GY(700)

COMMON/CBC/NODEBC(200),DIRBC(200),VALBC(200)

COMMON/CALT/ALT,L,H,Q,ALPHA

F(X)=(l.-(X/ALPHA)**2)*3.*VAV/2.

C

C GENERATE GEOMETRIC COORDINATES OF GLOBAL NODES:

READ(5,*) NX,NY,ALT,L,H,Q,ALPHA

ALPHA=ALPHA*3. 141592654/180.

RO=H/SIN(ALPHA)

DO J=1,NY+1THETAÂLPHA* ( 1 .

* ( J-l ) /NY )

XA=(L+R0)*(l.-COS(THETA) )

YA= ( L+RO ) *SIN ( THETA )

XB=(L+R0)-R0*COS( THETA)

YB=R0*S IN (THETA)

K=0

DO I=J,NX*(NY+1)+J,NY+1

GX( I ) =XA+ (XB-XA) * ( 1 . *K/NX)

GY(I)=YA+(YB-YA)*(1.*K/NX)

K=K+1

END DO

END DO

C

C GENERATE A MESH OF ELEMENTS AND CONNECTIVITY MATRIX:

READ(5,*) FX,FY

DO J=1,NX

T0P=J*(NY+1)+1

BOT=(J-D*(NY+l)+l

DO I=0,NY-1

ELEM=(J-D*NY+I+1

NODES ( ELEM , 1 ) =BOT+I

NODES ( ELEM , 2 ) =TOP+INODES ( ELEM , 3 ) =TOP+I+l

NODES ( ELEM , 4 ) =BOT+I+l

KIND(ELEM)=4

END DO

END DO

C

C SET BOUNDARY CONDITIONS ON ENTRANCE SIDE ACCORDING TO A PARABOLICC RADIAL VELOCITY DISTRIBUTION:

VAV=Q / ( (L+RO ) * 2 . *ALPHA )

A=0.

B=ALPHA*(1./NY)AREA=(B-A-(B*A3 + A**3) / (3.XAT.PHA**2) )*3.*VAV/2.

NODEBC(l)=l

DIRBC(1)=1

VALBC ( 1 ) =F (A ) +DELA

NODEBC(2)=l

DIRBC(2)=2

VALBC(2)=0.

DO J=1,NYTHETA=ALPHA* ( 1 .

*J/NY )

A=ALPHA* ( 1 .* ( J-l ) /NY )

AREA=((B-A) - (B**3 - A**3) /(3.*ALPHA**2) )*3.*VAV/2.

DELB=2*AREA/(B-A) -(F(A) + F(B)) - DELA

NODEBC(2*J+l)=J+l

DIRBC(2*J+1)=1

VALBC ( 2*J+l ) = ( F( B ) +DELB ) *COS ( THETA )

N0DEBC(2*J+2)=J+1

DIRBC(2*J+2)=2

VALBC ( 2*J+2 )-(F ( B ) +DELB ) *S IN ( THETA )

DELA=DELB

END DO

VALBC (2*NY+1)=0.

VALBC(2*NY+2)=0.

N0DE=NY+2

DO J=2,NX+1

BC=2*(NY+1) - 1 + J

NODEBC(BC)=NODE

DIRBC(BC)=2

VALBC (BC)=0.

NODE=NODE+NY+l

END DO

BC=2*(NY+1) + NX + 1

N0DE=2* ( NY+1 )

DO J=1,NX

NODEBC(BC)=NODE

NODEBC ( BC+1 ) =NODE

DIRBC(BC)=1

DIRBC(BC+1)=2

VALBC(BC)=0.

VALBC(BC)=0.

BC=BC+2

N0DE=N0DE+NY+1

END DO

C

RETURN

END

A*AA*AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAr

SUBROUTINE MESH9AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA*


C REGION.

INTEGER KIND,NODES,NODEBC,DIRBC,EX,EY,NX,NY,I,J,NODE,BC,ALT

REAL*8 FX,FY,GX,GY,VALBC,L,H,Q,ALPHA,R0,THETA,XA,YA,XB,YB,X,

+ F, CONST

COMMON /CELEM/KIND( 400 ) ,NODES (400,9 ) ,FX,FY

C0MM0N/CN0DE/GX(700),GY(700)

COMMON/ CBC /NODEBC (2 00 ),DIRBC( 200),VALBC (200)

COMMON/CALT/ALT,L ,H,Q ,ALPHA

F(X)=(l.-(SIN(X)/SIN(ALPHA))**2)*Q*SIN(ALPHA)**2./((L+R0)*CONST)

C


READ(5,*) EX,EY,ALT,L,H,Q,ALPHA

NX=2*EX+1

NY=2*EY+1

ALPHA=ALPHA*3. 141592654/180.

R0=H/SIN(ALPHA)

CONST=SIN(ALPHA)*COS(ALPHA) - ALPHA + 2*ALPHA*SIN(ALPHA)**2.

DO J=0,NY-1THETA=ALPHA* ( 1 .

*J / ( NY-1 ) )

XA= ( L+RO )*( 1 . -COS (THETA ) )

YA= ( L+RO ) *SIN ( THETA )

XB= (L+RO ) -R0*COS (THETA)

YB=R0*SIN( THETA)

DO I=0,NX-1GX(J+1+I*NY)=XA+(XB-XA)*(1.*I/(NX-1)>

GY(J+1+I*NY)=YA+(YB-YA)*(1.*I/(NX-D)

END DO

END DO

C


READ(5,*> FX,FY

DO 1=1,EX

DO J=1,EY

ELEM=(I-1)*EY + J

N0DE=(I-D*(2*NY) + 1 + 2*(J-1)

NODES ( ELEM , 1 ) =N0DE

NODES (ELEM, 2 )=N0DE


NODES ( ELEM, 4 ) =N0DE

NODES (ELEM, 5 )=N0DE



NODES ( ELEM, 8 )=N0DE


KIND (ELEM)* 9

END DO

END DO

r pttt BOUNDARY CONDITIONS ON ENTRANCE SIDE ACCORDING TO A PARABOLIC

C RADIAL VELOCITY DISTRIBUTION:

BC=1

DO J=0,NY-1

THETA=ALPHA* ( 1 . *J/ (NY-1 ) )

NODEBC (BC)=J+1

DIRBC(BC)=1

VALBC ( BC ) -F (THETA ) *COS ( THETA )

+ 2*NY

+ 2*NY +

+ 2

+ NY

+ 2*NY +

+ NY + 2

+ 1

+ NY + 1

NODEBC ( BC+1 ) *J+l

DIRBC(BC+1)=2VALBC ( BC+1 )= -F ( THETA ) *S IN ( THETA )

BC=BC+2END DO

DO J=2,NX

NODEBC ( BC )=( J - 1 ) *NY+1

DIRBC(BC)=2

VALBC(BC)=0.

BC=BC+1

END DO

DO J=2,NX

NODEBC (BC)=J*NY

NODEBC ( BC+1 )=J*NY

DIRBC(BC)=1

DIRBC(BC+1)=2

VALBC(BC)=0.

VALBC ( BC+1 )*0.

BC=BC+2

END DO

C

RETURN

ENDAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAr

SUBROUTINE MESHN02 *

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAr


C REGION. EX=11,14,17,20,... ONLY.

INTEGER KIND,NODES,NODEBC,DIRBC,EX,EY,NX,NY,I,J,NODE,MESH,+ BC,ALT,EN02

REAL*8 FX,FY,GX,GY,VALBC,L,H,Q,ALPHA,RO,THETA,XA,YA,XB,YB,VAV

C0MM0N/CELEM/KIND(400) ,NODES(400,9) ,FX,FY

COMMON/CNODE/GX(700),GY(700)

COMMON /CBC /NODEBC ( 200 ) ,DIRBC( 200 ) ,VALBC ( 200 )

COMMON/CALT/ALT ,L ,H , Q ,ALPHA ,EX ,EY

C


READ(5,*> EX,EY,ALT,L,H,Q,ALPHA

NX=2*EX+1

NY=2*EY+1

EN0Z=2*(EX-2)/3

ALPHA=ALPHA*3. 141592654/180.

R0=H/SIN (ALPHA)

DO J=0,NY-1

THETA=ALPHA* ( 1 .*J / ( NY-1 ) )

XA= ( L+RO ) * ( 1 . -COS (THETA) )

YA= ( L+RO ) *SIN (THETA )

XB= (L+RO ) -R0*COS( THETA)

YB=RO*SIN(THETA)

DO 1=0,1GX(I*NY+J+l)=(XA-L/4.) + ( (L/4. )-L/EN0Z)*I/2.

GY(I*NY+J+1)=YA

END DO

DO 1=2,3

GX(J+1+I*NY)=XA- 2.*L*(l.-I/4.)/EN02

GY(J+1+I*NY)=YA

END DO

DO I=4,2*EN0Z+4

GX(J+1+I*NY)=XA + (XB-XA)*(l.*(I-4)/(2.*EN0Z))

GY(J+1+I*NY)=YA + (YB-YA)*(l.*(I-4)/(2.*EN0Z))

END DO

DO I=2*EN0Z+5,NX-1

GX(J+1+I*NY)=XB + L*(I-2.*EN0Z-L/4.)/(2.*EN0Z)

GY(J+1+I*NY)=YBEND DO

END DO

C


READ(5,*) FX,FYDO 1=1,EX

DO J=1,EY

ELEM=(I-1)*EY + J

NODE=(I-l)*(2*NY) + 1 + 2*(J-1)

NODES ( ELEM , 1 ) =NODE

NODES (ELEM, 2 )=NODE + 2*NY

NODES (ELEM, 3 )=NODE + 2*NY + 2

NODES (ELEM, 4 )=N0DE + 2

NODES (ELEM, 5 )=NODE + NY

NODES (ELEM, 6 )=NODE + 2*NY + 1

NODES (ELEM, 7 )=NODE + NY + 2

NODES (ELEM, 8 )=N0DE + 1

NODES (ELEM, 9 )=NODE + NY + 1

KIND(ELEM)=9

END DO

END DO

C

C SET BOUNDARY CONDITIONS ON ENTRANCE SIDE ACCORDING TO A PARABOLIC,

C VELOCITY DISTRIBUTION:

VAV=Q/(2.*GY(5*NY))

BC=1

DO J=0,NY-1THETA=ALPHA* ( 1 . *J/ (NY-1 ) )

NODEBC (BCW+1

DIRBC ( BC ) = 1

VALBC(BC)=(l.-(GY(J+l)/GY(5*NY))**2.)*3.*VAV/2.

NODEBC ( BC+1 )=J+1

DIRBC (BC+1) =2

VALBC ( BC+1 )=0.

BC=BC+2

END DO

DO J=2,NX

NODEBC ( BC )=( J- 1 ) *NY+1

DIRBC (BC) =2

VALBC(BC)=0.

BC=BC+1

END DO

DO J=2,NX

NODEBC (BC)=J*NY

NODEBC ( BC+1 )=J*NY

DIRBC(BC)=1

DIRBC BC+1) =2

VALBC(BC)=0.

VALBC ( BC+1 )=0.

BC=BC+2

END DO

CRETURN

END*AAAAAAAAAAAAAAAAAAAAAAAAAAA******************************************************

SUBROUTINE PROC

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA**AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAC PROCESSOR SUBROUTINE. CALLS SUBROUTINES WHICH ASSEMBLE ELEMENT ANDC GLOBAL STIFFNESS MATRICES, APPLIES BOUNDARY CONDITIONS, SOLVES EQUATIONS.

CHARACTER*80 TITLE

INTEGER NNODE,NELEM,NBC,NPT,KIND,NODES ,HBAND,ELEM, I,J,NW,PRTREAL*B FX,FY,GK,GF,GU,EK,EFCOMMON /CCON /TITLE ,NNODE , NELEM ,NBC,NPTCOMMON/CELEM/KIND( 400 ) ,NODES ( 400 , 9 ) ,FX,FY

COMMON/CMATRX/HBAND,GU(1400),GK(1400,90),GF(1400),EK(1B,18),EF(18)COMMON/CPRINT/PRTdO)

C

C CALCULATE HALF-BANDWIDTH HBAND. FORMULA IS HBAND=(MAX DIF BETWEEN GLOBALC NODE NUMS. ON AN ELEMENT + 1)*(NUM. OF D.O.F. PER NODE)

HBAND=0

DO ELEM=1,NELEMDO I=1,KIND(ELEM)

DO J=1.KIND(ELEM)NW= ( IABS (NODES ( ELEM , I ) -NODES ( ELEM , J )) +1 ) *2

IF (NW .GT. HBAND) HBAND=NWEND DO

END DOEND DO

C

C INITIALIZE MATRICES:DO I=l,2*NNODE

GF(I)=0.

DO J=l,HBAND

GK(I,J)=0.

END DO

END DO

C

C CONSTRUCT ELEMENT AND GLOBAL MATRICES:

DO ELEM=1,NELEM

CALL ELEMENT ( ELEM,KIND( ELEM ) )

IF (PRT(l).EQ.l .AND. PRT(2) .LE.ELEM .AND. ELEM.LE.

+. PRTOJ) CALL SPILL (ELEM,KIND ( ELEM ),1)

. CALL ASSEMB(ELEM,KIND(ELEM))

END DO

C

IF (PRT(4) .EQ. 1) CALL SPILL(1,1,2)

CALL APLYBC

CALL GAUSSEL

C

RETURN

END

AAAAAAAAAAAAAAAAAAAAAAAAAAAA*AAA*AAAA***AA*****AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

SUBROUTINE ELEMENT ( ELEM,N)

AAAAAAAAAAAAAAAAAAAAAAAAAAAA********************AAA*AAAAAAAAAAAAAAAAAAAAAAAAAAAAA^

C ASSEMBLES THE ELEMENT STIFFNESS MATRIX.

INTEGER KIND,NODES,NINT,HBAND,ELEM, I,J,N,LXI,LNA

REAL*8 XI,NA,A,B,CPSI(9),DPSDX(9),DPSDY(9),DETJ,INTC,GU,+ FX,FY,MU,GAMMA,XIQ,WT,GK,GF,EK,EF

COMMON/CELEM/KIND( 400 ),NODES (400,9) ,FX,FY

C0MM0N/CINT/NINT,XIQ(5,5) ,WT(5,5)

COMM0N/CMATRX/HBAND,GU(1400),GK(14O0,90),GF(1400),EK(18,18),EF(18)

COMMON/CMAT/MU , GAMMA

C

C INITIALIZE ELEMENT MATRICES:

DO 1=1, 2*N

EF(I)=0.

DO J=1,2*N

EK(I,J)=0.

END DO

END DO

C

C SET DEGREE OF GAUSS INTEGRATION. LINEAR QUADS USE 2X2 POINTS,

IF (KIND(ELEM) .EQ. 4) NINT=2IF (KIND(ELEM) .NE. 4) NINT=3

C

C APPLY FULL INTEGRATION TO THE NON-PENALTY FUNCTION TERMS:

DO LXI=1,NINT

DO LNA=1,NINT

XI*XIQ(NINT,LXI)

NA=XIQ(NINT,LNA)

CALL SHAPE(ELEM,N,XI,NA,PSI,DPSDX,DPSDY,DETJ)

INTC=WT ( NINT ,LXI ) *WT (NINT , LNA ) *DETJ

DO 1=1,N

EF( I ) =EF( I )+FX*PSI ( I ) *INTC

EF( I+N ) =EF( I+N) +FY*PSI ( I ) *INTC

DO J=1,N

A=DPSDX ( I ) *DPSDX ( J ) * INTC

B=DPSDX ( J ) *DPSDY ( I ) *INTC

C=DPSDY(I)*DPSDY(J)*INTC

EK(I,J)=EK(I,J )+ 2.0*MU*A + MU*C

EK(I,J+N)=EK(I,J+N) + MU*B

EK(I+N,J+N)=EK(I+N,J+N) + MU*A + 2.*MU*C

END DO

END DO

END DO

END DO

C

C USE REDUCED INTEGRATION TO PENALTY FUNCTION TERMS:

DO LXI=1,NINT-1

DO LNA=1,NINT-1

XI=XIQ(NINT-1,LXI)

NA=XIQ(NINT-1,LNA)

CALL SHAPE(ELEM,N,XI,NA,PSI,DPSDX,DPSDY,DETJ)

INTC=WT(NINT-1 ,LXI ) *WT(NINT-1 ,LNA) *DETJ

DO 1=1,N

DO J-l-H

A=DPSDX ( I ) *DPSDX ( J ) * INTC

B=DPSDX ( I ) *DPSDY ( J ) *INTC

C=DPSDY ( I ) *DPSDY ( J ) *INTC

EK(I,J)=EK(I,J) + GAMMA*A

EK(I,J+N)=EK(I,J+N) + GAMMA*B

EK(I+N,J+N)=EK(I+N,J+N) + GAMMA*C

END DO

END DO

END DO

END DO

C

C CALCULATE LOWER LEFT TRANSPOSE PART OF CEK3 MATRIX:

DO 1=1,N

DO J=N+1,2*N

EK(J,I)=EK(I,J)

END DO

END DO

C

RETURN

END

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

SUBROUTINE SHAPE! ELEM,N,XI ,NA,PSI,DPSDX,DPSDY,DETJ)AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA*AAAAAAAAAAAA*AAAAAAAAAAAAAAAAAAAAAA

C AT GIVEN INTEGRATION POINTS IN LOCAL COORDINATES XI,NA , CALCULATE VALUE

C OF SHAPE FUNCTIONS, PARTIAL DERIVATIVES, AND TRANSFORM BACK INTO GLOBAL.

INTEGER N,ELEM

REAL*8 XI ,NA,PSI ( 9 ) ,DPSIDXI ( 9 ) ,DPSIDNA( 9 ) ,DPSDX( 9 ) ,DPSDY( 9 ) ,+ DETJ,JACI(2,2)

C

C CALCULATE VALUE OF SHAPE FUNCTIONS AND THEIR DERIVATIVES IN XI,NA COORDINAT

IF (N .EQ. 4) THEN

C FOUR NODE QUADRILATERAL: 4. . 2

PSI(1)*.25*(1-XI)*(1-NA)PSI ( 2 ) = .

25* ( 1+XI ) * ( 1-NA)

PSI(3)=.25*(1+XI)*(1+NA) 1. . <

PSI(4)=.25*(1-XI)*(1+NA)

DPSIDXI(1)=-.25*(1-NA)

DPSIDXI(2)=.25*(1-NA)

DPSIDXI(3)=.25*(1+NA)

DPSIDXI(4)=-.25*(1+NA)

DPSIDNA(1)=-.25*(1-XI)

DPSIDNA(2)=-.25*(1+XI)

DPSIDNA( 3 ) .25* ( 1+XI )

DPSIDNA(4)=.25*(1-XI)

ELSE IF (N .EQ. 8) THEN

C EIGHT NODE (SERENDIPITY) QUADRILATERAL:

PSI(1)=0.25*(1.-XI)*(1.-NA)*(-1.-XI-NA) 4. 7 I

PSI(2)=0.25*(1.+XI)*(1.-NA)*(-1.+XI-NA)

PSIO)=0.25*(1.+XI)*(1.+NA)*(-1.+XI+NA) 8 (

PSI ( 4 ) =0 .25* ( 1 . -XI ) * ( 1 . +NA) * ( -1 . -XI+NA)

PSI(5)=0.5*(1.-XI**2)*(1.-NA) 1. 5 .

PSI(6)=0.5*(1.+XI)*(1.-NA**2)

PSI(7)=0.5*(1.-XI**2)*(1.+NA)

PSI ( 8 ) =0 .5* ( 1 . -XI ) * ( 1 . -NA**2 )

DPSIDXI(1)=0.25*(1.-NA)*((-1.)*(-1.-XI-NA)+(1.-XI)*(-1.))

DPSIDXI ( 2 ) =0 .25* ( 1 . -NA) * ( ( 1 . ) * ( -1 . +XI-NA ) + ( 1 . +XI ) * ( 1 . ) )

DPSIDXI(3)=0.25*(1.+NA)*((1.)*(-1.+XI+NA)+(1.+XI)*(1.)>

DPSIDXI ( 4 ) =0. 25* ( 1 . +NA) * ( ( -1 . ) * ( -1 . -XI+NA)+( 1 . -XI ) * ( -1 . ) )

DPSIDXI ( 5 )XI* ( 1 . -NA)

DPSIDXI(6)=0.5*(1.-NA**2)

DPSIDXI ( 7 )XI* ( 1 . +NA)

DPSIDXI(B)=-0.5*(1.-NA**2)

DPSIDNA(1)=0.25*(1.-XI)*((-1.)*(-1.-XI-NA)+(1.-NA)*(-1.))

DPSIDNA(2)=0.25*(1.+XI)*((-1.)*(-1.+XI-NA)+(1.-NA)*(-1.))

DPSIDNA(3)=0.25*(1.+XI)*((1.)*(-1.+XI+NA)+(1.+NA)*(1.))

DPSIDNA(4)=0.25*(1.-XI)*((1.)*(-1.-XI+NA)+(1.+NA)*(1.))

DPSIDNA<5>0.5*(1.-XI**2)

DPSIDNA(6)=-(1.+XI)*NA

DPSIDNA(7)=0.5*(1.-XI**2)

DPSIDNA( 8 ) =- ( 1 . -XI ) *NA

ELSE IF (N .EQ. 9) THEN

C NINE NODE ISOPARAMETRIC QUADRILATERAL:

PSI(1)=0.25*(XI*XI-XI)*(NA*NA-NA) 4. 7 3

PSI ( 2) =0. 25* (XI*XI+XI )* (NA*NA-NA)

PSI(3)=0.25*(XI*XI+XI)*(NA*NA+NA) 8 9*

PSI(4)=0.25*(XI*XI-XI)*(NA*NA+NA)

PSI(5)=0.5*(1.-XI*XI)*(NA*NA-NA) 1 5 ;

PSI ( 6 ) =0 .5* (XI*XI+XI ) * ( 1 . -NA*NA )

PSI ( 7 ) =0 .5* ( 1 . -XI*XI ) * ( NA*NA+NA )

PSI(8)=0.5*(XI*XI-XI)*(1.-NA*NA)

PSI(9)=(1.-XI*XI)*(1.-NA*NA)

DPSIDXI(1)=0.25*(2.*XI-1.)*(NA*NA-NA)DPSIDXI ( 2 ) =0 .

25* ( 2 . *XI+1 . ) * (NA*NA-NA)

DPSIDXI(3)=0.25*(2.*XI+1.)*(NA*NA+NA)DPSIDXI(4)=0.25*(2.*XI-1.)*(NA*NA+NA)DPS IDXI ( 5 ) = -XI* (NA*NA-NA )

DPSIDXI(6)=0.5*(2.*XI+1)*(1.-NA*NA)DPSIDXI ( 7 ) =-XI* (NA*NA+NA)DPSIDXI ( 8 ) =0 .

5* ( 2 . *XI-1 . ) * ( 1 . -NA*NA~)

DPSIDXI ( 9 ) =-2 . *XI* ( 1 . -NA*NA)

DPSIDNA(1)=0.25*(XI*XI-XI)*(2.*NA-1.)

DPSIDNA(2)=0.25*(XI*XI+XI)*(2.*NA-1.)

DPSIDNAO)=0.25*(XI*XI+XI)*(2.*NA+1.)

DPSIDNA(4)=0.25*(XI*XI-XI)*(2.*NA+1.)DPSIDNA(5)=0.5*(1.-XI*XI)*(2.*NA-1.)DPSIDNA( 6) =- (XI*XI+XI ) *NA

DPS IDNA ( 7 ) = 0 .5* ( 1 . -XI*XI ) * ( 2 . *NA+1 . )

DPSIDNAt 8 ) =- (XI*XI-XI ) *NA

DPSIDNA(9)=-2.*(1.-XI*XI)*NAEND IF

C

C CALCULATE JACOBIAN TO TRANSFER PARTIAL DERIVATIVES FROM LOCAL TO GLOBAL:

CALL JACOBI (ELEM,N,DPSIDXI,DPS IDNA,DETJ,JACI)C CALCULATE PARTIAL DERIVATIVES OF SHAPE FUNCTIONS WITH RESPECT TO GLOBAL X,Y:

DO 1=1,N

DPSDX(I)=JACI(1,1)*DPSIDXI(I) + JACI(1,2)*DPSIDNA(I)

DPSDY(I)=JACI(2,1)*DPSIDXI(I) + JACI(2,2)*DPSIDNA(I)

END DO

C

RETURN

ENDAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAt

SUBROUTINE JACOBI (ELEM,N,DPSIDXI,DPSIDNA,DETJ,JACI)AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAr

C CALCULATE JACOBIAN MATRIX TO TRANSFORM FROM XI,NA TO X,Y.

INTEGER KIND,NODES, I,N,ELEM

REAL*8 DPSIDXI(9),DPSIDNA(9),DETJ,JACT(2,2),JACI(2,2),FX,FY,

+ GX,GY

COMMON/CNODE/GX( 700) ,GY(700)

COMMON/CELEM/KIND(400) ,N0DES(400,9) ,FX,FY

C

JACT(1,1)=0.

JACT(1,2)=0.

JACT(2,1)=0.

JACT(2,2)=0.

DO 1=1,NJACT(1,1)=JACT(1,1) + DPSIDXI(I)*GX(NODES(ELEM,D)

JACT(1,2)=JACT(1,2) + DPSIDXI ( I )*GY (NODES (ELEM, I ) )

JACT(2,1)=JACT(2,1) + DPSIDNA(I)*GX(NODES(ELEM,I) )

JACT(2,2)=JACT(2,2) + DPSIDNAt I ) *GY(NODES (ELEM, I ) )

END DO

C CALCULATE THE JACOBIAN DETERMINANT OF CJACT3:

DETJ=JACT(1,1)*JACT(2,2)- JACT(2,1)*JACT( 1,2)

IF (DETJ .LE. 0.0) THENWRITE(6,*)'

BAD JACOBIAN IN SUBROUTINE *JACOBI* '

STOP

END IF

C CALCULATE INVERSE OF CJACTD

JACI ( 1 , 1 ) =JACT( 2 , 2 ) /DETJ

JACI (1,2) =-JACT( 1,2) /DETJ

JACI (2,1) =-JACT( 2,1) /DETJ

JACI (2,2) =JACT( 1,1) /DETJC

RETURN

END

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA,

SUBROUTINE ASSEMB(ELEM,N)***********************AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAj

C ASSEMBLE GLOBAL MATRICES CGKD AND CGF3 FROM ELEMENTAL MATRICES CEKJ AND CEF

INTEGER KIND,NODES,HBAND, ELEM,N, I, J,UI,UJ,VI ,VJREAL*8 FX,FY,GK,GF,EK,EF,GUCOMMON/CELEM/KIND( 400 ) ,NODES (400,9) ,FX,FYCOMMON/CMATRX/HBAND,GU( 1400) ,GK(1400,90) ,GF(1400) ,EK(18,18) ,EF(18)

C THE FOLLOWING FOUR VARIABLES MARK THE POSITION IN GLOBAL MATRICES:

C UI=VERTICAL POSITION OF U VELOCITY

C VI=VERTICAL POSITION OF V VELOCITYC UJ=HORIZONTAL POSITION OF U VELOCITYC VJ=HORIZONTAL POSITION OF V VELOCITY

C ONLY SAVE UPPER TRIANGULAR PORTION OF CGKD. ENTRY GK(I,J) IS STORED

C IN BAND MATRIX AT POSITION GK(I,J-I+1).

DO 1=1,N

UI=NODES(ELEM,I)*2 - 1

VI=UI+1

GF(UI)=EF(I)

GF(VI)=EF(I+N)DO J=1,N

UJ=N0DES(ELEM,J)*2 - 1

VJ=UJ+1

GK(UI,UJ-UI+1)=GK(UI,UJ-UI+1)+EK(I,J)'

GK (UI ,VJ-UI+1 ) =GK (UI ,VJ-UI+1 ) +EK ( I , J+N)

GK(VI,UJ-VI+1)=GK(VI,UJ-VI+1)+EK(I+N,J)

GK(VI ,VJ-VI+1 ) =GK(VI ,VJ-VI+1 )+EK( I+N, J+N)

RETURN

ENDAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA**AAAAAAAAAAAAAAAA*

SUBROUTINE APLYBC

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAr

C APPLY BOUNDARY CONDITIONS AND POINT LOADS TO GLOBAL MATRICESCHARACTER*80 TITLE

INTEGER NNODE,NELEM,NBCNPT,NODEBCDIRBC,HBAND,NODEPT,DIRPT,BC,+ POS,POINT

REAL*8 BIG,VALBC,GK,GF,EK,EF,VALPT,GU

COMMON/CCON/TITLE,NNODE,NELEM,NBC,NPT

COMMON/CMATRX/HBAND,GU( 1400 ) ,GK( 1400, 90) ,GF( 1400) ,EK(18,18) ,EF(18)COMMON/CBC/N0DEBC(200),DIRBC(200),VALBC(200)

COMMON/CPT/NODEPT( 50 ) ,DIRPT( 50 ) ,VALPT( 50 )

C

C APPLY ESSENTIAL BOUNDARY CONDITION BY PENALTY METHOD:

DO BC=1,NBC

P0S=N0DEBC(BC)*2 - 2 + DIRBC (BC)

BIG=GK(P0S,1)*1.E10

GK(POS,l)=BIG

GF ( POS ) =BIG*VALBC ( BC )

END DO

C

C APPLY POINT LOADS:

DO POINT=l,NPT

IF (UI.LE.UJ)

IF (UI.LE.VJ)

IF (VI.LE.UJ)

IF (VI.LE.VJ)

END DO

END DO

P0S=N0DEPT(P0INT)*2 -2 + DIRPT(POINT)

GF(POS)=GF(POS) + VALPT(POINT)END DO

C

RETURN

END

*AAAAAAAAAAAAAAAAAAAAAAAA**AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

SUBROUTINE GAUSSEL******************* AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA*A******AAAAAAAAAAAA A

C SOLVE SYSTEM OF EQUATIONS OF THE FORM CGK3CGU3=CGF1 FOR CGU} USING

C GAUSS ELIMINATION. MATRIX CGKD IS STORED IN HALF-BAND FORM.

CHARACTER*80 TITLE

INTEGER NNODE,NELEM,NBC,NPT,HBAND,N,ROW,R,C,CMAX,J, JMAX

REAL*8 GK,GF,EK,EF,SUM,GU

COMMON /CCON /TITLE ,NNODE ,NELEM , NBC ,NPT

COMMON/CMATRX/HBAND,GU(1400) ,GK(1400,90) ,GF(1400) ,EK(18,18) ,EF(18)C

N=2*NNODE

C USE GAUSSIAN ELIMINATION TO GET AN UPPER TRIANGULAR MATRIX:

DO ROW=2,N

IF (ABS(GK(ROW-l,D) .LT. l.E-20) THENWRITE(6,*)'

GAUSS REDUCTION FAILED DUE TO SMALLPIVOT'

STOP

END IF

CMAX=MIN0 (HBAND- 1 ,N-ROW+l )

DO C=1,CMAX

SUM=0.

JMAX=MIN0 ( ROW-1 ,HBAND-C )

DO J=1,JMAX

SUM=SUM+GK(ROW-J,J+l)*GK(ROW-J,C+J)/GK(ROW-J,l)

END DO

GK(ROW,C)=GK(ROW,C)-SUM

END DO

END DO

C

C PERFORM FORWARD SUBSTITUTION OF CGF3:

DO R0W=2,N+1'

SUM=0.

JMAX=MIN0 (HBAND-1 ,ROW-l )

DO J=1,JMAX

SUM=SUM+GK(ROW-J,J+l)/GK(ROW-J,l)*GF(ROW-J)

END DO

GF(ROW) =GF(ROW) -SUM

END DO

C

C BACK SUBSTITUTE FOR CGU}:

GU(N)=GF(N)/GK(N,1)

DO R=2,N

ROW=N+l-R

CMAX=MIN0 (N,ROW+HBAND-l )

SUM=0.

DO C=ROW+l,CMAX

SUM=SUM + GU(C)*GK(ROW,C-ROW+l)

END DO

GU(ROW) = (GF(ROW)-SUM)/GK(ROW,1)

END DO

C

RETURN

END

AA*A*A*A***************************************************************AAAAAAAA

SUBROUTINE SPILL (ELEM,N, PRNT)*****************A******AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA>C SENDS MATRICES FOR OUTPUT (OR MODIFY TO OUTPUT TO A DATA FILE).

CHARACTER*80 TITLE

INTEGER NNODE,NELEM,NBC,NPT,HBAND,ELEM, PRNT, I,J,N,PRTREAL*8 GK,GF,EK,EF,GUCOMMON /CCON /TITLE ,NNODE , NELEM ,NBC,NPT

COMMON/CMATRX/HBAND,GU(1400),GK(1400,90),GF(1400),EK(18,18),EF(1B)COMMON /CPRINT /PRT ( 10 )

C PRNT=1 TO PRINT ELEMENT MATRIX; PRNT=2 TO PRINT GLOBAL MATRIX.C

IF (PRNT .EQ. 1) THEN

WRITE(6,20) ELEM

20 FORMATCO','

OUTPUT OF ELEMENT#',14,'

STIFFNESS MATRIX:')DO 1=1,N*2

WRITE(6,40) I40 FORMATC ', 'MATRIX ROW # ',14)

WRITE(6,60) (EK(I,J),J=1,N*2)60 F0RMAT(3(1X,6(E12.5,1X),/))

END DOWRITE(6,*)'

OUTPUT OF ELEMENT MATRIXCOMPLETED'

ELSE IF (PRNT .EQ. 2) THEN

WRITE(6,80)

80 F0RMAT('0'

, 'OUTPUT HALF-BAND GLOBAL STIFFNESS MATRIX:'

)DO I=PRT(5),PRT(6)

WRITE(6,100) I

100 FORMAT* IX, 'EQUATION # ',14)

WRITE(6,120) (GK(I,J),J=1,HBAND)

-120 FORMAT(30(1X,6(E12.5,1X),/))

END DOWRITE(6,*)'

OUTPUT OF GLOBAL MATRIXCOMPLETED'

END IF

C

RETURN

ENDAAAAAA AAAAAAAA****AA*A***A**AAA*AAAA*AA*AAAAAAAAA*A*A**AAAAAAAAAAAAAAAAAA AAAAAAAAA

SUBROUTINE POSTAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

INTEGER PRT

COMMON/CPRINT/PRTdO)

C OUTPUT FINAL NODAL SOLUTION:

IF (PRT(8) .EQ. 1) CALL PNODE

C CALCULATE PRESSURES AND OUTPUT FINAL ELEMENT SOLUTION:

CALL PELEM

WRITE(6,*) *********** END OF FINAL SOLUTION OUTPUT***********

C

RETURN

ENDAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAy

SUBROUTINE PNODE

A***A**************AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA;

C PRINT NODAL COORDINATES AND VELOCITY SOLUTIONS

CHARACTER*80 TITLE

INTEGER NNODE,NELEM,NBC,NPT,NODE,POS,HBAND

REAL*8 GX,GY,GU,GK,GF,EK,EF

COMMON/CCON/TITLE ,NNODE,NELEM ,NBC,NPT

COMMON/CNODE/GX( 700 ) ,GY( 700 )

COMMON/CMATRX/HBAND,GU( 1400) ,GK(1400,90) ,GF(1400) ,EK(18,18) ,EF(18)

CWRITE(6,20)

20 FORMAT('0','**********FINAL NODAL SOLUTION OUTPUT ***********

'

)

WRITE(6,30)

30FORMATCO','

NODE X-COOR Y-COOR U-VELOCITY

+ V-VELOCITY'

)

DO N0DE=1,NNODEP0S=2*N0DE - 1

WRITE(6,40) NODE,GX(NODE) ,GY(NODE) ,GU(POS) ,GU(P0S+1)

40 F0RMAT(1X,I4,5X,4(E12.5,4X))END DO

C

RETURN

END*A*A*AAAAAAAAAAAAA**AAAAAA***AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

SUBROUTINE PELEMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA***************************

C OUTPUT ELEMENT SOLUTION. PRESSURE IS COMPUTED AT GAUSS POINTS.

CHARACTER*80 TITLE

INTEGER NNODE,NELEM,NBC,NPT,KIND,NODES,NINT,ELEM,MAXITER,

+ LXI,LNA,I,POS,ALT,HBAND

REAL*8 FX,FY,GX,GY,XIQ,WT,MU,GAMMA,GU,GK,GF,EK,EF,XI,NA,PSI(9) ,+ DPSDX( 9 ) ,DPSDY( 9 ) ,DETJ ,X,Y,PRES ,PDROP ,R1 ,R2 ,L

COMMON /CCON /TITLE ,NNODE ,NELEM ,NBC,NPT

COMMON/CELEM/KIND(400) ,NODES(400,9) ,FX,FY

COMMON/CNODE/GX(700) ,GY(700)

C0MM0N/CINT/NINT,XIQ(5,5),WT(5,5)

COMMON/CMAT/MU,GAMMA

COMMON/CMATRX/HBAND,GU(1400),GK(1400,90),GF(1400),EK(18,1B),EF(18)

COMMON /CALT/ALT ,LC-

WRITE(6,20)

20 FORMAT('0'

,********** OUTPUT FINAL ELEMENT SOLUTION **********')

WRITE(6,30)

30 FORMAT('0'

, 'ELEMENT X-COOR Y-COOR PRESSURE')

PDROP=0.

R1 =0.

R2 =0.

DO ELEM=1,NELEM

IF (KIND(ELEM).EQ.4) NINT=2

IF (KIND(ELEM).GT.4) NINT=3

DO LXI=1,NINT-1

DO LNA=1,NINT-1

XI=XIQ(NINT-1,LXI)

NA=XIQ(NINT-1,LNA)

CALL SHAPE(ELEM,KIND(ELEM) ,XI,NA,PSI,DPSDX,

+ DPSDY,DETJ)

X=0.

Y=0.

PRES=0.

DO I=1,KIND(ELEM)

X=X+PS I ( I ) *GX ( NODES ( ELEM , I ) )

Y=Y+PSI ( I ) *GY (NODES ( ELEM , I ) )

POS=NODES (ELEM, I )*2-l

PRES=PRES+DPSDX ( I ) *GU ( POS ) +DPSDY ( I ) *GU ( POS+1 )

END DO

PRES = -GAMMA*PRES

IF (ALT .EQ. 2)

+CALL PERR0R(ELEM,LXI,PRES,PDR0P,X,Y,R1,R2)

WRITE(6,50) ELEM,X,Y,PRES

50F0RMAT(1X,I4,4X,3(E11.5,4X))

END DO

END DO

END DO

IF(ALT.EQ.2)WRITE(6,*) 'PRESSURE DROPIS'

,PDR0P*(R1-R2) /L

C

RETURN

END

*A**A**AAAAAAAAAAAAAAAAAAAAAAAA***AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA***

SUBROUTINE PERROR(ELEM,LXI,PRES,PDROP,X,Y,Rl,R2)AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

C COMPUTES PRESSURE DROP FOR NOZZLE PROBLEM WITH ENTRANCE AND EXIT.

INTEGER ELEM,LXI,EX,EY,ALT,ENOZREAL*8 PRES,PDR0P,X,Y,R,R0,R1,R2,L,H,Q,ALPHA

COMMON/CALT/ALT,L,H,Q,ALPHA,EX,EYC

EN0Z=2*(EX-2)/3

RO=H/SIN(ALPHA)

R=SQRT( (L+R0-X)**2. +Y**2.)

IF(ELEM.GE.EY+1.AND.ELEM.LE.2*EY.AND.LXI.EQ.2) THEN

PDROP=PDROP + PRES/(4.*EY)

R1=R1 + R/(4.*EY)END IF

IF(ELEM.GE.2*EY+1.AND. ELEM. LE.3*EY.AND.LXI.EQ.l) THEN

PDROP=PDROP + PRES/(4.*EY)

R1=R1 + R/(4.*EY)

END IF

IF(ELEM.GE.(EN0Z+1)*EY+1.AND.ELEM.LE.(EN0Z+2)*EY.AND.LXI.EQ.2)THEN

PDROP=PDROP - PRES/(4.*EY)

R2=R2 + R/(4.*EY)

END IF

IF(ELEM.GE. (ENOZ+2 ) *EY+1 .AND. ELEM. LE. (ENOZ+3 ) *EY.AND. LXI .EQ. 1 )THEN

PDROP=PDROP - PRES/(4.*EY)

R2=R2 + R/(4.*EY)

END IF

C

RETURN

END

Documents

An analysis of stroke fluid flow in a converging channel